Method and apparatus for video coding

ABSTRACT

Aspects of the disclosure provide methods and an apparatus for video encoding/decoding. The apparatus includes processing circuitry that decodes coded information of a coding unit (CU) from a coded video bitstream. The coded information indicates a last position of non-zero transform coefficients of a first coding block (CB) of the CU. The processing circuitry determines whether a secondary transform index is signaled in the coded information based on the last position and whether to perform a secondary transform on a second CB based on whether the secondary transform index is determined to be signaled in the coded information. When the secondary transform is determined to be performed, the processing circuitry performs the secondary transform on the second CB and reconstructs the second CB. Responsive to the secondary transform being determined not to be performed, the processing circuitry reconstructs the second CB without performing the secondary transform on the second CB.

INCORPORATION BY REFERENCE

This present application claims the benefit of priority to U.S.Provisional Application No. 62/829,435, “Modifications on the SecondaryTransform” filed on Apr. 4, 2019, which is incorporated by referenceherein in its entirety.

TECHNICAL FIELD

The present disclosure describes embodiments generally related to videocoding.

BACKGROUND

The background description provided herein is for the purpose ofgenerally presenting the context of the disclosure. Work of thepresently named inventors, to the extent the work is described in thisbackground section, as well as aspects of the description that may nototherwise qualify as prior art at the time of filing, are neitherexpressly nor impliedly admitted as prior art against the presentdisclosure.

Video coding and decoding can be performed using inter-pictureprediction with motion compensation. Uncompressed digital video caninclude a series of pictures, each picture having a spatial dimensionof, for example, 1920×1080 luminance samples and associated chrominancesamples. The series of pictures can have a fixed or variable picturerate (informally also known as frame rate), of, for example 60 picturesper second or 60 Hz. Uncompressed video has significant bitraterequirements. For example, 1080p60 4:2:0 video at 8 bit per sample(1920×1080 luminance sample resolution at 60 Hz frame rate) requiresclose to 1.5 Gbit/s bandwidth. An hour of such video requires more than600 GBytes of storage space.

One purpose of video coding and decoding can be the reduction ofredundancy in the input video signal, through compression. Compressioncan help reduce the aforementioned bandwidth or storage spacerequirements, in some cases by two orders of magnitude or more. Bothlossless and lossy compression, as well as a combination thereof can beemployed. Lossless compression refers to techniques where an exact copyof the original signal can be reconstructed from the compressed originalsignal. When using lossy compression, the reconstructed signal may notbe identical to the original signal, but the distortion between originaland reconstructed signals is small enough to make the reconstructedsignal useful for the intended application. In the case of video, lossycompression is widely employed. The amount of distortion tolerateddepends on the application; for example, users of certain consumerstreaming applications may tolerate higher distortion than users oftelevision distribution applications. The compression ratio achievablecan reflect that: higher allowable/tolerable distortion can yield highercompression ratios.

A video encoder and decoder can utilize techniques from several broadcategories, including, for example, motion compensation, transform,quantization, and entropy coding.

Video codec technologies can include techniques known as intra coding.In intra coding, sample values are represented without reference tosamples or other data from previously reconstructed reference pictures.In some video codecs, the picture is spatially subdivided into blocks ofsamples. When all blocks of samples are coded in intra mode, thatpicture can be an intra picture. Intra pictures and their derivationssuch as independent decoder refresh pictures, can be used to reset thedecoder state and can, therefore, be used as the first picture in acoded video bitstream and a video session, or as a still image. Thesamples of an intra block can be exposed to a transform, and thetransform coefficients can be quantized before entropy coding. Intraprediction can be a technique that minimizes sample values in thepre-transform domain. In some cases, the smaller the DC value after atransform is, and the smaller the AC coefficients are, the fewer thebits that are required at a given quantization step size to representthe block after entropy coding.

Traditional intra coding such as known from, for example MPEG-2generation coding technologies, does not use intra prediction. However,some newer video compression technologies include techniques thatattempt, from, for example, surrounding sample data and/or metadataobtained during the encoding/decoding of spatially neighboring, andpreceding in decoding order, blocks of data. Such techniques arehenceforth called “intra prediction” techniques. Note that in at leastsome cases, intra prediction is only using reference data from thecurrent picture under reconstruction and not from reference pictures.

There can be many different forms of intra prediction. When more thanone of such techniques can be used in a given video coding technology,the technique in use can be coded in an intra prediction mode. Incertain cases, modes can have submodes and/or parameters, and those canbe coded individually or included in the mode codeword. Which codewordto use for a given mode/submode/parameter combination can have an impactin the coding efficiency gain through intra prediction, and so can theentropy coding technology used to translate the codewords into abitstream.

A certain mode of intra prediction was introduced with H.264, refined inH.265, and further refined in newer coding technologies such as jointexploration model (JEM), versatile video coding (VVC), and benchmark set(BMS). A predictor block can be formed using neighboring sample valuesbelonging to already available samples. Sample values of neighboringsamples are copied into the predictor block according to a direction. Areference to the direction in use can be coded in the bitstream or mayitself be predicted.

Referring to FIG. 1A, depicted in the lower right is a subset of ninepredictor directions known from H.265's 33 possible predictor directions(corresponding to the 33 angular modes of the 35 intra modes). The pointwhere the arrows converge (101) represents the sample being predicted.The arrows represent the direction from which the sample is beingpredicted. For example, arrow (102) indicates that sample (101) ispredicted from a sample or samples to the upper right, at a 45 degreeangle from the horizontal. Similarly, arrow (103) indicates that sample(101) is predicted from a sample or samples to the lower left of sample(101), in a 22.5 degree angle from the horizontal.

Still referring to FIG. 1A, on the top left there is depicted a squareblock (104) of 4×4 samples (indicated by a dashed, boldface line). Thesquare block (104) includes 16 samples, each labelled with an “S”, itsposition in the Y dimension (e.g., row index) and its position in the Xdimension (e.g., column index). For example, sample S21 is the secondsample in the Y dimension (from the top) and the first (from the left)sample in the X dimension. Similarly, sample S44 is the fourth sample inblock (104) in both the Y and X dimensions. As the block is 4×4 samplesin size, S44 is at the bottom right. Further shown are reference samplesthat follow a similar numbering scheme. A reference sample is labelledwith an R, its Y position (e.g., row index) and X position (columnindex) relative to block (104). In both H.264 and H.265, predictionsamples neighbor the block under reconstruction; therefore no negativevalues need to be used.

Intra picture prediction can work by copying reference sample valuesfrom the neighboring samples as appropriated by the signaled predictiondirection. For example, assume the coded video bitstream includessignaling that, for this block, indicates a prediction directionconsistent with arrow (102)—that is, samples are predicted from aprediction sample or samples to the upper right, at a 45 degree anglefrom the horizontal. In that case, samples S41, S32, S23, and S14 arepredicted from the same reference sample R05. Sample S44 is thenpredicted from reference sample R08.

In certain cases, the values of multiple reference samples may becombined, for example through interpolation, in order to calculate areference sample; especially when the directions are not evenlydivisible by 45 degrees.

The number of possible directions has increased as video codingtechnology has developed. In H.264 (year 2003), nine different directioncould be represented. That increased to 33 in H.265 (year 2013), andJEM/VVC/BMS, at the time of disclosure, can support up to 65 directions.Experiments have been conducted to identify the most likely directions,and certain techniques in the entropy coding are used to represent thoselikely directions in a small number of bits, accepting a certain penaltyfor less likely directions. Further, the directions themselves cansometimes be predicted from neighboring directions used in neighboring,already decoded, blocks.

FIG. 1B shows a schematic (180) that depicts 65 intra predictiondirections according to JEM to illustrate the increasing number ofprediction directions over time.

The mapping of intra prediction directions bits in the coded videobitstream that represent the direction can be different from videocoding technology to video coding technology; and can range, forexample, from simple direct mappings of prediction direction to intraprediction mode, to codewords, to complex adaptive schemes involvingmost probable modes, and similar techniques. In all cases, however,there can be certain directions that are statistically less likely tooccur in video content than certain other directions. As the goal ofvideo compression is the reduction of redundancy, those less likelydirections will, in a well working video coding technology, berepresented by a larger number of bits than more likely directions.

Video coding and decoding can be performed using inter-pictureprediction with motion compensation. Motion compensation can be a lossycompression technique and can relate to techniques where a block ofsample data from a previously reconstructed picture or part thereof(reference picture), after being spatially shifted in a directionindicated by a motion vector (MV henceforth), is used for the predictionof a newly reconstructed picture or picture part. In some cases, thereference picture can be the same as the picture currently underreconstruction. MVs can have two dimensions X and Y, or threedimensions, the third being an indication of the reference picture inuse (the latter, indirectly, can be a time dimension).

In some video compression techniques, an MV applicable to a certain areaof sample data can be predicted from other MVs, for example from thoserelated to another area of sample data spatially adjacent to the areaunder reconstruction, and preceding that MV in decoding order. Doing socan substantially reduce the amount of data required for coding the MV,thereby removing redundancy and increasing compression. MV predictioncan work effectively, for example, because when coding an input videosignal derived from a camera (known as natural video) there is astatistical likelihood that areas larger than the area to which a singleMV is applicable move in a similar direction and, therefore, can in somecases be predicted using a similar motion vector derived from MVs ofneighboring area. That results in the MV found for a given area to besimilar or the same as the MV predicted from the surrounding MVs, andthat in turn can be represented, after entropy coding, in a smallernumber of bits than what would be used if coding the MV directly. Insome cases, MV prediction can be an example of lossless compression of asignal (namely: the MVs) derived from the original signal (namely: thesample stream). In other cases, MV prediction itself can be lossy, forexample because of rounding errors when calculating a predictor fromseveral surrounding MVs.

Various MV prediction mechanisms are described in H.265/HEVC (ITU-T Rec.H.265, “High Efficiency Video Coding”, December 2016). Out of the manyMV prediction mechanisms that H.265 offers, described here is atechnique henceforth referred to as “spatial merge”.

Referring to FIG. 2, a current block (201) comprises samples that havebeen found by the encoder during the motion search process to bepredictable from a previous block of the same size that has beenspatially shifted. Instead of coding that MV directly, the MV can bederived from metadata associated with one or more reference pictures,for example from the most recent (in decoding order) reference picture,using the MV associated with either one of five surrounding samples,denoted A0, A1, and B0, B1, B2 (202 through 206, respectively). InH.265, the MV prediction can use predictors from the same referencepicture that the neighboring block is using.

SUMMARY

Aspects of the disclosure provide methods and apparatuses for videoencoding/decoding. In some examples, an apparatus for video decodingincludes processing circuitry. The processing circuitry can decode codedinformation of a coding unit (CU) from a coded video bitstream. Thecoded information can indicate a last position of non-zero transformcoefficients of a first coding block (CB) of the CU. The processingcircuitry can determine whether a secondary transform index is signaledin the coded information based on the last position. The processingcircuitry can determine whether to perform a secondary transform on asecond CB based on whether the secondary transform index is determinedto be signaled in the coded information. Responsive to the secondarytransform being determined to be performed, the processing circuitry canperform the secondary transform on the second CB and reconstructing thesecond CB. Responsive to the secondary transform being determined not tobe performed, the processing circuitry can reconstruct the second CBwithout performing the secondary transform on the second CB.

In an embodiment, the processing circuitry can determine whether ahorizontal component of the last position is less then a first thresholdand a vertical component of the last position is less than a secondthreshold. Responsive to the horizontal component being determined to beless than the first threshold and the vertical component beingdetermined to be less than the second threshold, the processingcircuitry can determine that the secondary transform index is notsignaled in the coded information.

In an embodiment, the processing circuitry can determine whether a sumof a horizontal component and a vertical component of the last positionis less than a threshold. Responsive to the sum being determined to beless than the threshold, the processing circuitry can determine that thesecondary transform index is not signaled in the coded information.

In an embodiment, the processing circuitry can determine whether aminimum one of (i) a horizontal component and (ii) a vertical componentof the last position is less than a threshold. Responsive to the minimumone being determined to be less than the threshold, the processingcircuitry can determine that the secondary transform index is notsignaled in the coded information.

In an embodiment, the processing circuitry can determine whether amaximum one of (i) a horizontal component and (ii) a vertical componentof the last position is less than a threshold. Responsive to the maximumone being determined to be less than the threshold, the processingcircuitry can determine that the secondary transform index is notsignaled in the coded information.

In an embodiment, the first CB is a luma block. The last position is alast luma position for the luma block. The processing circuitry candetermine whether the second transform index is signaled based on thelast luma position.

In an embodiment, the first CB is a luma block. The last position is alast luma position for the luma block. The CU further includes a chromablock. The coding information further indicates a last chroma positionof non-zero transform coefficients for the chroma block. The processingcircuitry can determine whether the secondary transform index issignaled based on the last luma position and the last chroma position.

In some examples, an apparatus for video decoding includes processingcircuitry. The processing circuitry can decode coding information of acoding unit (CU) from a coded video bitstream. The coding informationcan indicate a size of the CU. The processing circuitry can determinewhether a secondary transform is allowed based on the size of the CU anda CU size threshold where when the size of the CU is less than or equalto the CU size threshold, the secondary transform is determined to beallowed, and when the size of the CU is larger than the CU sizethreshold, the secondary transform is determined not to be allowed.

In an embodiment, the CU size threshold is a maximum size of a transformunit in the CU.

In an embodiment, when the size of the CU is less than or equal to theCU size threshold, the processing circuitry can determine a number ofnon-zero transform coefficients for at least one CB in the CU where asize of each of the at least one CB is larger than or equal to a firstthreshold. Responsive to the number of non-zero transform coefficientsbeing less than a second threshold, the processing circuitry candetermine that a secondary transform index is not signaled in the codedinformation. In an example, the coding information indicates that acolor format for the CU is YUV 4:2:0. The CU includes a luma block andtwo chroma blocks. The processing circuitry can determine whether afirst dimension of the luma block is 4 and a second dimension of theluma block is N where N is larger than or equal to 4. Responsive to thefirst and second dimensions being determined to be 4 and N,respectively, the processing circuitry can determine the number ofnon-zero transform coefficients from only the luma block where the atleast one CB is the luma block.

In an embodiment, the coding information indicates that a color formatfor the CU is YUV 4:2:2. The CU includes a luma block and two chromablocks. The processing circuitry can determine whether a size of theluma block is 4×N where N is larger than or equal to 4. Responsive tothe size of the luma block being determined to be 4×N where N and 4 area height and a width of the luma block, respectively, the processingcircuitry can determine the number of non-zero transform coefficientsfrom only the luma block. The at least one CB is the luma block.

Aspects of the disclosure also provide a non-transitorycomputer-readable medium storing instructions which when executed by acomputer for video decoding cause the computer to perform any of themethods for video decoding.

BRIEF DESCRIPTION OF THE DRAWINGS

Further features, the nature, and various advantages of the disclosedsubject matter will be more apparent from the following detaileddescription and the accompanying drawings in which:

FIG. 1A is a schematic illustration of an exemplary subset of intraprediction modes.

FIG. 1B is an illustration of exemplary intra prediction directions.

FIG. 2 is a schematic illustration of a current block and itssurrounding spatial merge candidates in one example.

FIG. 3 is a schematic illustration of a simplified block diagram of acommunication system (300) in accordance with an embodiment.

FIG. 4 is a schematic illustration of a simplified block diagram of acommunication system (400) in accordance with an embodiment.

FIG. 5 is a schematic illustration of a simplified block diagram of adecoder in accordance with an embodiment.

FIG. 6 is a schematic illustration of a simplified block diagram of anencoder in accordance with an embodiment.

FIG. 7 shows a block diagram of an encoder in accordance with anotherembodiment.

FIG. 8 shows a block diagram of a decoder in accordance with anotherembodiment.

FIG. 9 shows an example of transform unit syntax in accordance with anembodiment.

FIGS. 10A-10C show an example of residual coding syntax in accordancewith an embodiment.

FIGS. 11A-11B show examples of primary transforms in accordance with anembodiment.

FIGS. 12A-12E show an example of transformation process in accordancewith an embodiment.

FIG. 13 shows an exemplary transform coding process (1300).

FIG. 14 shows an exemplary transform coding process (1400).

FIG. 15A shows an exemplary process (1501) of a reduced forwardtransform and an exemplary process (1502) of a reduced inversetransform.

FIGS. 15B-15C show examples of reduced secondary transforms inaccordance with some embodiments.

FIG. 15D shows an exemplary transform set selection able (1550).

FIG. 16A shows an illustration of exemplary intra prediction directionsand the intra prediction modes in accordance with an embodiment.

FIG. 16B shows an illustration of exemplary intra prediction directionsand the corresponding intra prediction modes in accordance with anembodiment.

FIG. 17 shows an example of 4 reference lines for a coding block (1710)in accordance with an embodiment.

FIG. 18 shows a table that associates a number of sub-partitions with ablock size in accordance with an embodiment.

FIG. 19 shows an example of sub-partitions of a block having a size 4×8or 8×4.

FIG. 20 shows another example of sub-partitions of a block having a sizelarger than 4×8 and 8×4.

FIGS. 21A-21D show examples of different YUV formats.

FIG. 22 shows a flow chart outlining a process (2200) in accordance withan embodiment.

FIG. 23 shows a flow chart outlining a process (2300) in accordance withan embodiment.

FIG. 24 is a schematic illustration of a computer system in accordancewith an embodiment.

DETAILED DESCRIPTION OF EMBODIMENTS

FIG. 3 illustrates a simplified block diagram of a communication system(300) according to an embodiment of the present disclosure. Thecommunication system (300) includes a plurality of terminal devices thatcan communicate with each other, via, for example, a network (350). Forexample, the communication system (300) includes a first pair ofterminal devices (310) and (320) interconnected via the network (350).In the FIG. 3 example, the first pair of terminal devices (310) and(320) performs unidirectional transmission of data. For example, theterminal device (310) may code video data (e.g., a stream of videopictures that are captured by the terminal device (310)) fortransmission to the other terminal device (320) via the network (350).The encoded video data can be transmitted in the form of one or morecoded video bitstreams. The terminal device (320) may receive the codedvideo data from the network (350), decode the coded video data torecover the video pictures and display video pictures according to therecovered video data. Unidirectional data transmission may be common inmedia serving applications and the like.

In another example, the communication system (300) includes a secondpair of terminal devices (330) and (340) that performs bidirectionaltransmission of coded video data that may occur, for example, duringvideoconferencing. For bidirectional transmission of data, in anexample, each terminal device of the terminal devices (330) and (340)may code video data (e.g., a stream of video pictures that are capturedby the terminal device) for transmission to the other terminal device ofthe terminal devices (330) and (340) via the network (350). Eachterminal device of the terminal devices (330) and (340) also may receivethe coded video data transmitted by the other terminal device of theterminal devices (330) and (340), and may decode the coded video data torecover the video pictures and may display video pictures at anaccessible display device according to the recovered video data.

In the FIG. 3 example, the terminal devices (310), (320), (330) and(340) may be illustrated as servers, personal computers and smart phonesbut the principles of the present disclosure may be not so limited.Embodiments of the present disclosure find application with laptopcomputers, tablet computers, media players and/or dedicated videoconferencing equipment. The network (350) represents any number ofnetworks that convey coded video data among the terminal devices (310),(320), (330) and (340), including for example wireline (wired) and/orwireless communication networks. The communication network (350) mayexchange data in circuit-switched and/or packet-switched channels.Representative networks include telecommunications networks, local areanetworks, wide area networks and/or the Internet. For the purposes ofthe present discussion, the architecture and topology of the network(350) may be immaterial to the operation of the present disclosureunless explained herein below.

FIG. 4 illustrates, as an example for an application for the disclosedsubject matter, the placement of a video encoder and a video decoder ina streaming environment. The disclosed subject matter can be equallyapplicable to other video enabled applications, including, for example,video conferencing, digital TV, storing of compressed video on digitalmedia including CD, DVD, memory stick and the like, and so on.

A streaming system may include a capture subsystem (413), that caninclude a video source (401), for example a digital camera, creating forexample a stream of video pictures (402) that are uncompressed. In anexample, the stream of video pictures (402) includes samples that aretaken by the digital camera. The stream of video pictures (402),depicted as a bold line to emphasize a high data volume when compared toencoded video data (404) (or coded video bitstreams), can be processedby an electronic device (420) that includes a video encoder (403)coupled to the video source (401). The video encoder (403) can includehardware, software, or a combination thereof to enable or implementaspects of the disclosed subject matter as described in more detailbelow. The encoded video data (404) (or encoded video bitstream (404)),depicted as a thin line to emphasize the lower data volume when comparedto the stream of video pictures (402), can be stored on a streamingserver (405) for future use. One or more streaming client subsystems,such as client subsystems (406) and (408) in FIG. 4 can access thestreaming server (405) to retrieve copies (407) and (409) of the encodedvideo data (404). A client subsystem (406) can include a video decoder(410), for example, in an electronic device (430). The video decoder(410) decodes the incoming copy (407) of the encoded video data andcreates an outgoing stream of video pictures (411) that can be renderedon a display (412) (e.g., display screen) or other rendering device (notdepicted). In some streaming systems, the encoded video data (404),(407), and (409) (e.g., video bitstreams) can be encoded according tocertain video coding/compression standards. Examples of those standardsinclude ITU-T Recommendation H.265. In an example, a video codingstandard under development is informally known as Versatile Video Coding(VVC). The disclosed subject matter may be used in the context of VVC.

It is noted that the electronic devices (420) and (430) can includeother components (not shown). For example, the electronic device (420)can include a video decoder (not shown) and the electronic device (430)can include a video encoder (not shown) as well.

FIG. 5 shows a block diagram of a video decoder (510) according to anembodiment of the present disclosure. The video decoder (510) can beincluded in an electronic device (530). The electronic device (530) caninclude a receiver (531) (e.g., receiving circuitry). The video decoder(510) can be used in the place of the video decoder (410) in the FIG. 4example.

The receiver (531) may receive one or more coded video sequences to bedecoded by the video decoder (510); in the same or another embodiment,one coded video sequence at a time, where the decoding of each codedvideo sequence is independent from other coded video sequences. Thecoded video sequence may be received from a channel (501), which may bea hardware/software link to a storage device which stores the encodedvideo data. The receiver (531) may receive the encoded video data withother data, for example, coded audio data and/or ancillary data streams,that may be forwarded to their respective using entities (not depicted).The receiver (531) may separate the coded video sequence from the otherdata. To combat network jitter, a buffer memory (515) may be coupled inbetween the receiver (531) and an entropy decoder/parser (520) (“parser(520)” henceforth). In certain applications, the buffer memory (515) ispart of the video decoder (510). In others, it can be outside of thevideo decoder (510) (not depicted). In still others, there can be abuffer memory (not depicted) outside of the video decoder (510), forexample to combat network jitter, and in addition another buffer memory(515) inside the video decoder (510), for example to handle playouttiming. When the receiver (531) is receiving data from a store/forwarddevice of sufficient bandwidth and controllability, or from anisosynchronous network, the buffer memory (515) may not be needed, orcan be small. For use on best effort packet networks such as theInternet, the buffer memory (515) may be required, can be comparativelylarge and can be advantageously of adaptive size, and may at leastpartially be implemented in an operating system or similar elements (notdepicted) outside of the video decoder (510).

The video decoder (510) may include the parser (520) to reconstructsymbols (521) from the coded video sequence. Categories of those symbolsinclude information used to manage operation of the video decoder (510),and potentially information to control a rendering device such as arender device (512) (e.g., a display screen) that is not an integralpart of the electronic device (530) but can be coupled to the electronicdevice (530), as was shown in FIG. 5. The control information for therendering device(s) may be in the form of Supplemental EnhancementInformation (SEI messages) or Video Usability Information (VUI)parameter set fragments (not depicted). The parser (520) mayparse/entropy-decode the coded video sequence that is received. Thecoding of the coded video sequence can be in accordance with a videocoding technology or standard, and can follow various principles,including variable length coding, Huffman coding, arithmetic coding withor without context sensitivity, and so forth. The parser (520) mayextract from the coded video sequence, a set of subgroup parameters forat least one of the subgroups of pixels in the video decoder, based uponat least one parameter corresponding to the group. Subgroups can includeGroups of Pictures (GOPs), pictures, tiles, slices, macroblocks, CodingUnits (CUs), blocks, Transform Units (TUs), Prediction Units (PUs) andso forth. The parser (520) may also extract from the coded videosequence information such as transform coefficients, quantizer parametervalues, motion vectors, and so forth.

The parser (520) may perform an entropy decoding/parsing operation onthe video sequence received from the buffer memory (515), so as tocreate symbols (521).

Reconstruction of the symbols (521) can involve multiple different unitsdepending on the type of the coded video picture or parts thereof (suchas: inter and intra picture, inter and intra block), and other factors.Which units are involved, and how, can be controlled by the subgroupcontrol information that was parsed from the coded video sequence by theparser (520). The flow of such subgroup control information between theparser (520) and the multiple units below is not depicted for clarity.

Beyond the functional blocks already mentioned, the video decoder (510)can be conceptually subdivided into a number of functional units asdescribed below. In a practical implementation operating undercommercial constraints, many of these units interact closely with eachother and can, at least partly, be integrated into each other. However,for the purpose of describing the disclosed subject matter, theconceptual subdivision into the functional units below is appropriate.

A first unit is the scaler/inverse transform unit (551). Thescaler/inverse transform unit (551) receives a quantized transformcoefficient as well as control information, including which transform touse, block size, quantization factor, quantization scaling matrices,etc. as symbol(s) (521) from the parser (520). The scaler/inversetransform unit (551) can output blocks comprising sample values, thatcan be input into aggregator (555).

In some cases, the output samples of the scaler/inverse transform (551)can pertain to an intra coded block; that is: a block that is not usingpredictive information from previously reconstructed pictures, but canuse predictive information from previously reconstructed parts of thecurrent picture. Such predictive information can be provided by an intrapicture prediction unit (552). In some cases, the intra pictureprediction unit (552) generates a block of the same size and shape ofthe block under reconstruction, using surrounding already reconstructedinformation fetched from the current picture buffer (558). The currentpicture buffer (558) buffers, for example, partly reconstructed currentpicture and/or fully reconstructed current picture. The aggregator(555), in some cases, adds, on a per sample basis, the predictioninformation the intra prediction unit (552) has generated to the outputsample information as provided by the scaler/inverse transform unit(551).

In other cases, the output samples of the scaler/inverse transform unit(551) can pertain to an inter coded, and potentially motion compensatedblock. In such a case, a motion compensation prediction unit (553) canaccess reference picture memory (557) to fetch samples used forprediction. After motion compensating the fetched samples in accordancewith the symbols (521) pertaining to the block, these samples can beadded by the aggregator (555) to the output of the scaler/inversetransform unit (551) (in this case called the residual samples orresidual signal) so as to generate output sample information. Theaddresses within the reference picture memory (557) from where themotion compensation prediction unit (553) fetches prediction samples canbe controlled by motion vectors, available to the motion compensationprediction unit (553) in the form of symbols (521) that can have, forexample X, Y, and reference picture components. Motion compensation alsocan include interpolation of sample values as fetched from the referencepicture memory (557) when sub-sample exact motion vectors are in use,motion vector prediction mechanisms, and so forth.

The output samples of the aggregator (555) can be subject to variousloop filtering techniques in the loop filter unit (556). Videocompression technologies can include in-loop filter technologies thatare controlled by parameters included in the coded video sequence (alsoreferred to as coded video bitstream) and made available to the loopfilter unit (556) as symbols (521) from the parser (520), but can alsobe responsive to meta-information obtained during the decoding ofprevious (in decoding order) parts of the coded picture or coded videosequence, as well as responsive to previously reconstructed andloop-filtered sample values.

The output of the loop filter unit (556) can be a sample stream that canbe output to the render device (512) as well as stored in the referencepicture memory (557) for use in future inter-picture prediction.

Certain coded pictures, once fully reconstructed, can be used asreference pictures for future prediction. For example, once a codedpicture corresponding to a current picture is fully reconstructed andthe coded picture has been identified as a reference picture (by, forexample, the parser (520)), the current picture buffer (558) can becomea part of the reference picture memory (557), and a fresh currentpicture buffer can be reallocated before commencing the reconstructionof the following coded picture.

The video decoder (510) may perform decoding operations according to apredetermined video compression technology in a standard, such as ITU-TRec. H.265. The coded video sequence may conform to a syntax specifiedby the video compression technology or standard being used, in the sensethat the coded video sequence adheres to both the syntax of the videocompression technology or standard and the profiles as documented in thevideo compression technology or standard. Specifically, a profile canselect certain tools as the only tools available for use under thatprofile from all the tools available in the video compression technologyor standard. Also necessary for compliance can be that the complexity ofthe coded video sequence is within bounds as defined by the level of thevideo compression technology or standard. In some cases, levels restrictthe maximum picture size, maximum frame rate, maximum reconstructionsample rate (measured in, for example megasamples per second), maximumreference picture size, and so on. Limits set by levels can, in somecases, be further restricted through Hypothetical Reference Decoder(HRD) specifications and metadata for HRD buffer management signaled inthe coded video sequence.

In an embodiment, the receiver (531) may receive additional (redundant)data with the encoded video. The additional data may be included as partof the coded video sequence(s). The additional data may be used by thevideo decoder (510) to properly decode the data and/or to moreaccurately reconstruct the original video data. Additional data can bein the form of, for example, temporal, spatial, or signal noise ratio(SNR) enhancement layers, redundant slices, redundant pictures, forwarderror correction codes, and so on.

FIG. 6 shows a block diagram of a video encoder (603) according to anembodiment of the present disclosure. The video encoder (603) isincluded in an electronic device (620). The electronic device (620)includes a transmitter (640) (e.g., transmitting circuitry). The videoencoder (603) can be used in the place of the video encoder (403) in theFIG. 4 example.

The video encoder (603) may receive video samples from a video source(601) (that is not part of the electronic device (620) in the FIG. 6example) that may capture video image(s) to be coded by the videoencoder (603). In another example, the video source (601) is a part ofthe electronic device (620).

The video source (601) may provide the source video sequence to be codedby the video encoder (603) in the form of a digital video sample streamthat can be of any suitable bit depth (for example: 8 bit, 10 bit, 12bit, . . . ), any colorspace (for example, BT.601 Y CrCB, RGB, . . . ),and any suitable sampling structure (for example Y CrCb 4:2:0, Y CrCb4:4:4). In a media serving system, the video source (601) may be astorage device storing previously prepared video. In a videoconferencingsystem, the video source (601) may be a camera that captures local imageinformation as a video sequence. Video data may be provided as aplurality of individual pictures that impart motion when viewed insequence. The pictures themselves may be organized as a spatial array ofpixels, wherein each pixel can comprise one or more samples depending onthe sampling structure, color space, etc. in use. A person skilled inthe art can readily understand the relationship between pixels andsamples. The description below focuses on samples.

According to an embodiment, the video encoder (603) may code andcompress the pictures of the source video sequence into a coded videosequence (643) in real time or under any other time constraints asrequired by the application. Enforcing appropriate coding speed is onefunction of a controller (650). In some embodiments, the controller(650) controls other functional units as described below and isfunctionally coupled to the other functional units. The coupling is notdepicted for clarity. Parameters set by the controller (650) can includerate control related parameters (picture skip, quantizer, lambda valueof rate-distortion optimization techniques, . . . ), picture size, groupof pictures (GOP) layout, maximum motion vector search range, and soforth. The controller (650) can be configured to have other suitablefunctions that pertain to the video encoder (603) optimized for acertain system design.

In some embodiments, the video encoder (603) is configured to operate ina coding loop. As an oversimplified description, in an example, thecoding loop can include a source coder (630) (e.g., responsible forcreating symbols, such as a symbol stream, based on an input picture tobe coded, and a reference picture(s)), and a (local) decoder (633)embedded in the video encoder (603). The decoder (633) reconstructs thesymbols to create the sample data in a similar manner as a (remote)decoder also would create (as any compression between symbols and codedvideo bitstream is lossless in the video compression technologiesconsidered in the disclosed subject matter). The reconstructed samplestream (sample data) is input to the reference picture memory (634). Asthe decoding of a symbol stream leads to bit-exact results independentof decoder location (local or remote), the content in the referencepicture memory (634) is also bit exact between the local encoder andremote encoder. In other words, the prediction part of an encoder “sees”as reference picture samples exactly the same sample values as a decoderwould “see” when using prediction during decoding. This fundamentalprinciple of reference picture synchronicity (and resulting drift, ifsynchronicity cannot be maintained, for example because of channelerrors) is used in some related arts as well.

The operation of the “local” decoder (633) can be the same as of a“remote” decoder, such as the video decoder (510), which has alreadybeen described in detail above in conjunction with FIG. 5. Brieflyreferring also to FIG. 5, however, as symbols are available andencoding/decoding of symbols to a coded video sequence by an entropycoder (645) and the parser (520) can be lossless, the entropy decodingparts of the video decoder (510), including the buffer memory (515), andparser (520) may not be fully implemented in the local decoder (633).

An observation that can be made at this point is that any decodertechnology except the parsing/entropy decoding that is present in adecoder also necessarily needs to be present, in substantially identicalfunctional form, in a corresponding encoder. For this reason, thedisclosed subject matter focuses on decoder operation. The descriptionof encoder technologies can be abbreviated as they are the inverse ofthe comprehensively described decoder technologies. Only in certainareas a more detail description is required and provided below.

During operation, in some examples, the source coder (630) may performmotion compensated predictive coding, which codes an input picturepredictively with reference to one or more previously-coded picture fromthe video sequence that were designated as “reference pictures”. In thismanner, the coding engine (632) codes differences between pixel blocksof an input picture and pixel blocks of reference picture(s) that may beselected as prediction reference(s) to the input picture.

The local video decoder (633) may decode coded video data of picturesthat may be designated as reference pictures, based on symbols createdby the source coder (630). Operations of the coding engine (632) mayadvantageously be lossy processes. When the coded video data may bedecoded at a video decoder (not shown in FIG. 6), the reconstructedvideo sequence typically may be a replica of the source video sequencewith some errors. The local video decoder (633) replicates decodingprocesses that may be performed by the video decoder on referencepictures and may cause reconstructed reference pictures to be stored inthe reference picture cache (634). In this manner, the video encoder(603) may store copies of reconstructed reference pictures locally thathave common content as the reconstructed reference pictures that will beobtained by a far-end video decoder (absent transmission errors).

The predictor (635) may perform prediction searches for the codingengine (632). That is, for a new picture to be coded, the predictor(635) may search the reference picture memory (634) for sample data (ascandidate reference pixel blocks) or certain metadata such as referencepicture motion vectors, block shapes, and so on, that may serve as anappropriate prediction reference for the new pictures. The predictor(635) may operate on a sample block-by-pixel block basis to findappropriate prediction references. In some cases, as determined bysearch results obtained by the predictor (635), an input picture mayhave prediction references drawn from multiple reference pictures storedin the reference picture memory (634).

The controller (650) may manage coding operations of the source coder(630), including, for example, setting of parameters and subgroupparameters used for encoding the video data.

Output of all aforementioned functional units may be subjected toentropy coding in the entropy coder (645). The entropy coder (645)translates the symbols as generated by the various functional units intoa coded video sequence, by lossless compressing the symbols according totechnologies such as Huffman coding, variable length coding, arithmeticcoding, and so forth.

The transmitter (640) may buffer the coded video sequence(s) as createdby the entropy coder (645) to prepare for transmission via acommunication channel (660), which may be a hardware/software link to astorage device which would store the encoded video data. The transmitter(640) may merge coded video data from the video coder (603) with otherdata to be transmitted, for example, coded audio data and/or ancillarydata streams (sources not shown).

The controller (650) may manage operation of the video encoder (603).During coding, the controller (650) may assign to each coded picture acertain coded picture type, which may affect the coding techniques thatmay be applied to the respective picture. For example, pictures oftenmay be assigned as one of the following picture types:

An Intra Picture (I picture) may be one that may be coded and decodedwithout using any other picture in the sequence as a source ofprediction. Some video codecs allow for different types of intrapictures, including, for example Independent Decoder Refresh (“IDR”)Pictures. A person skilled in the art is aware of those variants of Ipictures and their respective applications and features.

A predictive picture (P picture) may be one that may be coded anddecoded using intra prediction or inter prediction using at most onemotion vector and reference index to predict the sample values of eachblock.

A bi-directionally predictive picture (B Picture) may be one that may becoded and decoded using intra prediction or inter prediction using atmost two motion vectors and reference indices to predict the samplevalues of each block. Similarly, multiple-predictive pictures can usemore than two reference pictures and associated metadata for thereconstruction of a single block.

Source pictures commonly may be subdivided spatially into a plurality ofsample blocks (for example, blocks of 4×4, 8×8, 4×8, or 16×16 sampleseach) and coded on a block-by-block basis. Blocks may be codedpredictively with reference to other (already coded) blocks asdetermined by the coding assignment applied to the blocks' respectivepictures. For example, blocks of I pictures may be codednon-predictively or they may be coded predictively with reference toalready coded blocks of the same picture (spatial prediction or intraprediction). Pixel blocks of P pictures may be coded predictively, viaspatial prediction or via temporal prediction with reference to onepreviously coded reference picture. Blocks of B pictures may be codedpredictively, via spatial prediction or via temporal prediction withreference to one or two previously coded reference pictures.

The video encoder (603) may perform coding operations according to apredetermined video coding technology or standard, such as ITU-T Rec.H.265. In its operation, the video encoder (603) may perform variouscompression operations, including predictive coding operations thatexploit temporal and spatial redundancies in the input video sequence.The coded video data, therefore, may conform to a syntax specified bythe video coding technology or standard being used.

In an embodiment, the transmitter (640) may transmit additional datawith the encoded video. The source coder (630) may include such data aspart of the coded video sequence. Additional data may comprisetemporal/spatial/SNR enhancement layers, other forms of redundant datasuch as redundant pictures and slices, SEI messages, VUI parameter setfragments, and so on.

A video may be captured as a plurality of source pictures (videopictures) in a temporal sequence. Intra-picture prediction (oftenabbreviated to intra prediction) makes use of spatial correlation in agiven picture, and inter-picture prediction makes uses of the (temporalor other) correlation between the pictures. In an example, a specificpicture under encoding/decoding, which is referred to as a currentpicture, is partitioned into blocks. When a block in the current pictureis similar to a reference block in a previously coded and still bufferedreference picture in the video, the block in the current picture can becoded by a vector that is referred to as a motion vector. The motionvector points to the reference block in the reference picture, and canhave a third dimension identifying the reference picture, in casemultiple reference pictures are in use.

In some embodiments, a bi-prediction technique can be used in theinter-picture prediction. According to the bi-prediction technique, tworeference pictures, such as a first reference picture and a secondreference picture that are both prior in decoding order to the currentpicture in the video (but may be in the past and future, respectively,in display order) are used. A block in the current picture can be codedby a first motion vector that points to a first reference block in thefirst reference picture, and a second motion vector that points to asecond reference block in the second reference picture. The block can bepredicted by a combination of the first reference block and the secondreference block.

Further, a merge mode technique can be used in the inter-pictureprediction to improve coding efficiency.

According to some embodiments of the disclosure, predictions, such asinter-picture predictions and intra-picture predictions are performed inthe unit of blocks. For example, according to the HEVC standard, apicture in a sequence of video pictures is partitioned into coding treeunits (CTU) for compression, the CTUs in a picture have the same size,such as 64×64 pixels, 32×32 pixels, or 16×16 pixels. In general, a CTUincludes three coding tree blocks (CTBs), which are one luma CTB and twochroma CTBs. Each CTU can be recursively quadtree split into one ormultiple coding units (CUs). For example, a CTU of 64×64 pixels can besplit into one CU of 64×64 pixels, or 4 CUs of 32×32 pixels, or 16 CUsof 16×16 pixels. In an example, each CU is analyzed to determine aprediction type for the CU, such as an inter prediction type or an intraprediction type. The CU is split into one or more prediction units (PUs)depending on the temporal and/or spatial predictability. Generally, eachPU includes a luma prediction block (PB), and two chroma PBs. In anembodiment, a prediction operation in coding (encoding/decoding) isperformed in the unit of a prediction block. Using a luma predictionblock as an example of a prediction block, the prediction block includesa matrix of values (e.g., luma values) for pixels, such as 8×8 pixels,16×16 pixels, 8×16 pixels, 16×8 pixels, and the like.

FIG. 7 shows a diagram of a video encoder (703) according to anotherembodiment of the disclosure. The video encoder (703) is configured toreceive a processing block (e.g., a prediction block) of sample valueswithin a current video picture in a sequence of video pictures, andencode the processing block into a coded picture that is part of a codedvideo sequence. In an example, the video encoder (703) is used in theplace of the video encoder (403) in the FIG. 4 example.

In an HEVC example, the video encoder (703) receives a matrix of samplevalues for a processing block, such as a prediction block of 8×8samples, and the like. The video encoder (703) determines whether theprocessing block is best coded using intra mode, inter mode, orbi-prediction mode using, for example, rate-distortion optimization.When the processing block is to be coded in intra mode, the videoencoder (703) may use an intra prediction technique to encode theprocessing block into the coded picture; and when the processing blockis to be coded in inter mode or bi-prediction mode, the video encoder(703) may use an inter prediction or bi-prediction technique,respectively, to encode the processing block into the coded picture. Incertain video coding technologies, merge mode can be an inter pictureprediction submode where the motion vector is derived from one or moremotion vector predictors without the benefit of a coded motion vectorcomponent outside the predictors. In certain other video codingtechnologies, a motion vector component applicable to the subject blockmay be present. In an example, the video encoder (703) includes othercomponents, such as a mode decision module (not shown) to determine themode of the processing blocks.

In the FIG. 7 example, the video encoder (703) includes the interencoder (730), an intra encoder (722), a residue calculator (723), aswitch (726), a residue encoder (724), a general controller (721), andan entropy encoder (725) coupled together as shown in FIG. 7.

The inter encoder (730) is configured to receive the samples of thecurrent block (e.g., a processing block), compare the block to one ormore reference blocks in reference pictures (e.g., blocks in previouspictures and later pictures), generate inter prediction information(e.g., description of redundant information according to inter encodingtechnique, motion vectors, merge mode information), and calculate interprediction results (e.g., predicted block) based on the inter predictioninformation using any suitable technique. In some examples, thereference pictures are decoded reference pictures that are decoded basedon the encoded video information.

The intra encoder (722) is configured to receive the samples of thecurrent block (e.g., a processing block), in some cases compare theblock to blocks already coded in the same picture, generate quantizedcoefficients after transform, and in some cases also intra predictioninformation (e.g., an intra prediction direction information accordingto one or more intra encoding techniques). In an example, the intraencoder (722) also calculates intra prediction results (e.g., predictedblock) based on the intra prediction information and reference blocks inthe same picture.

The general controller (721) is configured to determine general controldata and control other components of the video encoder (703) based onthe general control data. In an example, the general controller (721)determines the mode of the block, and provides a control signal to theswitch (726) based on the mode. For example, when the mode is the intramode, the general controller (721) controls the switch (726) to selectthe intra mode result for use by the residue calculator (723), andcontrols the entropy encoder (725) to select the intra predictioninformation and include the intra prediction information in thebitstream; and when the mode is the inter mode, the general controller(721) controls the switch (726) to select the inter prediction resultfor use by the residue calculator (723), and controls the entropyencoder (725) to select the inter prediction information and include theinter prediction information in the bitstream.

The residue calculator (723) is configured to calculate a difference(residue data) between the received block and prediction resultsselected from the intra encoder (722) or the inter encoder (730). Theresidue encoder (724) is configured to operate based on the residue datato encode the residue data to generate the transform coefficients. In anexample, the residue encoder (724) is configured to convert the residuedata from a spatial domain to a frequency domain, and generate thetransform coefficients. The transform coefficients are then subject toquantization processing to obtain quantized transform coefficients. Invarious embodiments, the video encoder (703) also includes a residuedecoder (728). The residue decoder (728) is configured to performinverse-transform, and generate the decoded residue data. The decodedresidue data can be suitably used by the intra encoder (722) and theinter encoder (730). For example, the inter encoder (730) can generatedecoded blocks based on the decoded residue data and inter predictioninformation, and the intra encoder (722) can generate decoded blocksbased on the decoded residue data and the intra prediction information.The decoded blocks are suitably processed to generate decoded picturesand the decoded pictures can be buffered in a memory circuit (not shown)and used as reference pictures in some examples.

The entropy encoder (725) is configured to format the bitstream toinclude the encoded block. The entropy encoder (725) is configured toinclude various information according to a suitable standard, such asthe HEVC standard. In an example, the entropy encoder (725) isconfigured to include the general control data, the selected predictioninformation (e.g., intra prediction information or inter predictioninformation), the residue information, and other suitable information inthe bitstream. Note that, according to the disclosed subject matter,when coding a block in the merge submode of either inter mode orbi-prediction mode, there is no residue information.

FIG. 8 shows a diagram of a video decoder (810) according to anotherembodiment of the disclosure. The video decoder (810) is configured toreceive coded pictures that are part of a coded video sequence, anddecode the coded pictures to generate reconstructed pictures. In anexample, the video decoder (810) is used in the place of the videodecoder (410) in the FIG. 4 example.

In the FIG. 8 example, the video decoder (810) includes an entropydecoder (871), an inter decoder (880), a residue decoder (873), areconstruction module (874), and an intra decoder (872) coupled togetheras shown in FIG. 8.

The entropy decoder (871) can be configured to reconstruct, from thecoded picture, certain symbols that represent the syntax elements ofwhich the coded picture is made up. Such symbols can include, forexample, the mode in which a block is coded (such as, for example, intramode, inter mode, bi-predicted mode, the latter two in merge submode oranother submode), prediction information (such as, for example, intraprediction information or inter prediction information) that canidentify certain sample or metadata that is used for prediction by theintra decoder (872) or the inter decoder (880), respectively, residualinformation in the form of, for example, quantized transformcoefficients, and the like. In an example, when the prediction mode isinter or bi-predicted mode, the inter prediction information is providedto the inter decoder (880); and when the prediction type is the intraprediction type, the intra prediction information is provided to theintra decoder (872). The residual information can be subject to inversequantization and is provided to the residue decoder (873).

The inter decoder (880) is configured to receive the inter predictioninformation, and generate inter prediction results based on the interprediction information.

The intra decoder (872) is configured to receive the intra predictioninformation, and generate prediction results based on the intraprediction information.

The residue decoder (873) is configured to perform inverse quantizationto extract de-quantized transform coefficients, and process thede-quantized transform coefficients to convert the residual from thefrequency domain to the spatial domain. The residue decoder (873) mayalso require certain control information (to include the QuantizerParameter (QP)), and that information may be provided by the entropydecoder (871) (data path not depicted as this may be low volume controlinformation only).

The reconstruction module (874) is configured to combine, in the spatialdomain, the residual as output by the residue decoder (873) and theprediction results (as output by the inter or intra prediction modulesas the case may be) to form a reconstructed block, that may be part ofthe reconstructed picture, which in turn may be part of thereconstructed video. It is noted that other suitable operations, such asa deblocking operation and the like, can be performed to improve thevisual quality.

It is noted that the video encoders (403), (603), and (703), and thevideo decoders (410), (510), and (810) can be implemented using anysuitable technique. In an embodiment, the video encoders (403), (603),and (703), and the video decoders (410), (510), and (810) can beimplemented using one or more integrated circuits. In anotherembodiment, the video encoders (403), (603), and (603), and the videodecoders (410), (510), and (810) can be implemented using one or moreprocessors that execute software instructions.

Aspects of the disclosure are related to modifications on a secondarytransform, such as implementations of an inverse secondary transform.

In some embodiments, such as in HEVC, a primary transform may include4-point, 8-point, 16-point and 32-point discrete cosine transform (DCT)type 2 (DCT-2), and the transform core matrices may be represented using8-bit integers (i.e., 8-bit transform core). The transform core matricesof a smaller DCT-2 are part of transform core matrices of a largerDCT-2, as shown in APPENDIX I.

The DCT-2 core matrices show symmetry/anti-symmetry characteristics.Therefore, a “partial butterfly” implementation may be supported toreduce a number of operation counts (e.g., multiplications, additions,subtractions, shifts, and/or the like), and identical results of matrixmultiplication can be obtained using the partial butterfly.

In some embodiments, such as in VVC, besides the 4-point, 8-point,16-point, and 32-point DCT-2 transforms described above, additional2-point and 64-point DCT-2 may also be included. An example of a64-point DCT-2 core, such as used in VVC, is shown in APPENDIX II as a64×64 matrix.

In addition to DCT-2 and 4×4 DST-7, such as used in HEVC, an AdaptiveMultiple Transform (AMT) (also known as Enhanced Multiple Transform(EMT) or Multiple Transform Selection (MTS)) scheme, can be used, suchas in VVC, for residual coding for both inter and intra coded blocks.The AMT scheme may use multiple selected transforms from the DCT/DSTfamilies other than the current transforms in HEVC. The newly introducedtransform matrices are DST-7 and DCT-8. Table 1 shows examples of thebasis functions of the selected DST/DCT for an N-point input.

TABLE 1 Transform Type Basis function T_(i)(j), i, j = 0, 1, . . . , N −1 DCT-2${T_{i}(j)} = {{\omega_{0} \cdot \sqrt{\frac{2}{N}} \cdot \cos}\mspace{11mu} \left( \frac{\pi \cdot i \cdot \left( {{2j} + 1} \right)}{2N} \right)}$${{where}\mspace{14mu} \omega_{0}} = \left\{ \begin{matrix}\sqrt{\frac{2}{N}} & {i = 0} \\1 & {i \neq 0}\end{matrix} \right.$ DCT-8${T_{i}(j)} = {{\sqrt{\frac{4}{{2N} + 1}} \cdot \cos}\mspace{11mu} \left( \frac{\pi \cdot \left( {{2i} + 1} \right) \cdot \left( {{2j} + 1} \right)}{{4N} + 2} \right)}$DST-7${T_{i}(j)} = {{\sqrt{\frac{4}{{2N} + 1}} \cdot \sin}\mspace{11mu} \left( \frac{\pi \cdot \left( {{2i} + 1} \right) \cdot \left( {j + 1} \right)}{{2N} + 1} \right)}$

The primary transform matrices, such as used in VVC, may be used with8-bit representation. The AMT applies transform matrices to the CUs withboth a width and a height smaller than or equal to 32. Whether AMT isapplied may be controlled by a flag (e.g., an mts_flag). When themts_flag is equal to 0, in some examples, only DCT-2 is applied forcoding residue data. When the mts_flag is equal to 1, an index (e.g., anmts_idx) may be further signalled using 2 bins to identify thehorizontal and vertical transform to be used according to Table 2, wherea type value of 1 means DST-7 is used, and a type value of 2 means DCT-8is used. In Table 2, the specification of trTypeHor and trTypeVerdepends on mts_idx[x][y][cIdx].

TABLE 2 mts_idx[xTbY][yTbY][cIdx] trTypeHor trTypeVer −1 0 0 0 1 1 1 2 12 1 2 3 2 2

In some embodiments, an implicit MTS can be applied when the abovesignaling based MTS (i.e., explicit MTS) is not used. With the implicitMTS, the transform selection is made according to the block width andheight instead of the signaling. For example, with an implicit MTS,DST-7 is selected for a shorter side (i.e., a minimum one of M and N) ofthe block of M×N and DCT-2 is selected for a longer side (i.e., amaximum one of M and N) of the block.

Exemplary transform cores, each of which is a matrix composed by thebasis vectors, of DST-7 and DCT-8 are illustrated in APPENDIX III.

In some examples, such as in VVC, when both the height and width of thecoding block is smaller than or equal to 64, the TB size is the same asthe coding block size. When either the height or width of the codingblock is larger than 64, when doing a transform (such as an inversetransform, an inverse primary transform, or the like) or intraprediction, the coding block is further split into multiple sub-blocks,where the width and height of each sub-block is smaller than or equal to64. One transform can be performed on each sub-block.

Related syntax and semantics of MTS in some examples in VVC can bedescribed below (highlighted using gray color) in FIGS. 9 and 10A-10C.FIG. 9 shows an example of transform unit syntax. FIGS. 10A-10C show anexample of a residual coding syntax.

An example of the transform unit semantics is as follows.cu_mts_flag[x0][y0]equal to 1 specifies that multiple transformselection is applied to the residual samples of the associated lumatransform block. cu_mts_flag[x0][y0] equal to 0 specifies that multipletransform selection is not applied to the residual samples of theassociated luma transform block. The array indices x0, y0 specify thelocation (x0, y0) of the top-left luma sample of the consideredtransform block relative to the top-left luma sample of the picture.When cu_mts_flag[x0][y0] is not present, it is inferred to be equal to0.

An example of the residual coding semantics is as follows.mts_idx[x0][y0] specifies which transform kernels are applied to theluma residual samples along the horizontal and vertical direction of thecurrent transform block. The array indices x0, y0 specify the location(x0, y0) of the top-left luma sample of the considered transform blockrelative to the top-left luma sample of the picture. Whenmts_idx[x0][y0] is not present, it is inferred to be equal to −1.

FIG. 11A shows an exemplary forward transform (also referred to as aforward primary transform) performed by an encoder. The forwardtransform can include a forward horizontal transform and a forwardvertical transform. The forward horizontal transform is applied first toa residual block (1110) having residual data to obtain an intermediateblock. Subsequently, the forward vertical transform is applied to theintermediate block to obtain a coefficient block (1112) having transformcoefficients.

FIG. 11B shows an exemplary backward transform (also referred to as aninverse primary transform or an inverse transform) performed by adecoder. Generally speaking, the inverse transform matches the forwardtransform. The inverse primary transform can include an inverse primaryhorizontal transform (also referred to as an inverse horizontaltransform) and an inverse primary vertical transform (also referred toas an inverse vertical transform). To match the forward transform, anorder of applying the inverse horizontal and vertical transforms isswitched in the inverse transform. Accordingly, the inverse verticaltransform is applied first to a coefficient block (1122) to obtain anintermediate block. Subsequently, the inverse horizontal transform isapplied to the intermediate block to obtain a residual block (1120).

A primary transform can refer to a forward primary transform or aninverse primary transform. A horizontal transform can refer to aninverse horizontal transform or a forward horizontal transform.Similarly, a vertical transform can refer to an inverse verticaltransform or a forward vertical transform.

In an example, such as in VVC, at the decoder, the inverse verticalprimary transform is performed first, then the inverse horizontalprimary transform is performed second after applying the inversevertical transform, as indicated in FIGS. 12A-12E in texts highlightedin gray color. FIGS. 12A-12E show an example of a transformationprocess, for example, for scaled transform coefficients. The textshighlighted in gray color are shown FIG. 12E.

In an embodiment, a mode-dependent non-separable secondary transform(NSST) can be used between a forward core transform and a quantizationat an encoder side and between a de-quantization and an inverse coretransform at a decoder side. For example, to keep a low complexity, aNSST is applied to low frequency coefficients after a primary transform(or a core transform). When both a width (W) and a height (H) of atransform coefficient block are larger than or equal to 8, an 8×8 NSSTis applied to a top-left 8×8 region of the transform coefficients block.Otherwise, when either the width W or the height H of the transformcoefficient block is 4, a 4×4 NSST is applied, and the 4×4 NSST isperformed on a top-left min(8,W)×min(8,H) region of the transformcoefficient block. The above transform selection method is applied forboth luma and chroma components.

A matrix multiplication implementation of a NSST is described as followsusing a 4×4 input block as an example. The 4×4 input block X is writtenin Eq. (1) as

$\begin{matrix}{X = \begin{bmatrix}X_{00} & X_{01} & X_{02} & X_{03} \\X_{10} & X_{11} & X_{12} & X_{13} \\X_{20} & X_{21} & X_{22} & X_{23} \\X_{30} & X_{31} & X_{32} & X_{33}\end{bmatrix}} & (1)\end{matrix}$

The input block X can be represented as a vector

in Eq. (2) where

{right arrow over (X)}=[X ₀₀ X ₀₁ X ₀₂ X ₀₃ X ₁₀ X ₁₁ X ₁₂ X ₁₃ X ₂₀ X₂₁ X ₂₂ X ₂₃ X ₃₀ X ₃₁ X ₃₂ X ₃₃]^(T)  (2)

The non-separable transform is calculated as

=T·

, where

indicates a transform coefficient vector, and T is a 16×16 transformmatrix. The 16×1 transform coefficient vector

is subsequently reorganized as a 4×4 block using a scanning order (forexample, a horizontal scanning order, a vertical scanning order or adiagonal scanning order) for the input block X. Coefficients withsmaller indices can be placed with smaller scanning indices in the 4×4coefficient block. In some embodiments, a Hypercube-Givens Transform(HyGT) with a butterfly implementation can be used instead of the matrixmultiplication described above to reduce the complexity of the NSST.

In an example, 35×3 non-separable secondary transforms are available forboth 4×4 and 8×8 block sizes, where 35 is a number of transform setsassociated with the intra prediction modes, and 3 is a number of NSSTcandidates for each intra prediction mode. Table 3 shows an exemplarymapping from an intra prediction mode to a respective transform set. Atransform set applied to luma/chroma transform coefficients is specifiedby a corresponding luma/chroma intra prediction mode, according to Table3 that shows mapping from an intra prediction mode to a transform setindex. For an intra prediction mode larger than 34, which corresponds toa diagonal prediction direction, a transform coefficient block istransposed before/after the NSST at the encoder/decoder, respectively.

For each transform set, a selected NSST candidate can be furtherspecified by an explicitly signaled CU level NSST index. The CU levelNSST index is signaled in a bitstream for each intra coded CU aftertransform coefficients and a truncated unary binarization is used forthe CU level NSST index. For example, a truncated value is 2 for theplanar or the DC mode, and 3 for an angular intra prediction mode. In anexample, the CU level NSST index is signaled only when there is morethan one non-zero coefficient in the CU. The default value is zero andnot signaled, indicating that a NS ST is not applied to the CU. Each ofvalues 1-3 indicates which NSST candidate is to be applied from thetransform set.

TABLE 3 Intra mode 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 set 0 1 2 34 5 6 7 8 9 10 11 12 13 14 15 16 Intra mode 17 18 19 20 21 22 23 24 2526 27 28 29 30 31 32 33 set 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3132 33 Intra mode 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 set34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 Intra mode 51 52 5354 55 56 57 58 59 60 61 62 63 64 65 66 67(LM) set 17 16 15 14 13 12 1110 9 8 7 6 5 4 3 2 NULL

In some embodiments, a NSST is not applied for a block coded with atransform skip mode. When the CU level NSST index is signaled for a CUand not equal to zero, a NSST is not used for a block that is coded withthe transform skip mode in the CU. When the CU with blocks of allcomponents are coded in a transform skip mode or a number of non-zerocoefficients of non-transform-skip mode CBs is less than 2, the CU levelNSST index is not signaled for the CU.

In some embodiments, a variant of NSST, such as a reduced size transform(RST), is employed. The RST uses a transform zero-out scheme. In anexample, in the RST, whether the intra prediction mode is Planar or DCis checked for entropy coding the transform index of NSST.

In an example, 4 transform sets are applied, and each transform setincludes three RST transform cores. The RST transform cores can have asize of 16×48 (or 16×64) (applied for transform coefficient block with aheight and width both being greater than or equal to 8) or 16×16(applied for transform coefficient block with either height or widthbeing equal to 4). For notational convenience, the 16×48 (or 16×64)transform is denoted as RST8×8 and the 16×16 one as RST4×4.

FIG. 13 and FIG. 14 show examples of two transform coding processes(1300) and (1400) using a 16×64 transform (or a 64×16 transformdepending on whether the transform is a forward or inverse secondarytransform) and a 16×48 transform (or a 48×16 transform depending onwhether the transform is a forward or inverse secondary transform),respectively. Referring to FIG. 13, in the process (1300), at an encoderside, a forward primary transform (1310) can first be performed over aresidual block to obtain a coefficient block (1313). Subsequently, aforward secondary transform (1312) can be applied to the coefficientblock (1313). In the forward secondary transform (1312), 64 coefficientsof 4×4 sub-blocks A-D at a top-left corner of the coefficient block(1313) can be represented by a 64-length vector, and the 64-lengthvector can be multiplied with a transform matrix of 64×16 (i.e., a widthof 64 and a height of 16) according to Eq.(2), resulting in a 16-lengthvector. Elements in the 16-length vector are filled back into thetop-left 4×4 sub-block A of the coefficient block (1313). Thecoefficients in the sub-blocks B-D can be zero. The resultingcoefficients after the forward secondary transform (1312) are thenquantized at a step (1314), and entropy-coded to generate coded bits ina bitstream (1316).

The coded bits can be received at a decoder side, and entropy-decodedfollowed by a de-quantization (1324) to generate a coefficient block(1323). An inverse secondary transform (1322), such as an inverse RST8×8can be performed to obtain 64 coefficients, for example, from the 16coefficients at a top-left 4×4 sub-block E. The 64 coefficients can befilled back to the 4×4 sub-blocks E-H. Further, the coefficients in thecoefficient block (1323) after the inverse secondary transform (1322)can be processed with an inverse primary transform (1320) to obtain arecovered residual block.

The process (1400) of the FIG. 14 example is similar to the process(1300) except that fewer (i.e., 48) coefficients are processed duringthe forward secondary transform (1412). Specifically, the 48coefficients in the sub-blocks A-C are processed with a smallertransform matrix of a size of 48×16. Using the smaller transform matrixof 48×16 can reduce a memory size for storing the transform matrix and anumber of calculations (e.g., multiplications, additions, subtractions,and/or the like), and thus can reduce computation complexity.

A Reduced Transform (RT) (also referred to as RST) can map an Ndimensional vector to an R dimensional vector in a different space,where R/N (R<N) is a reduction factor.

The RST (or RT) matrix is an R×N matrix as follows:

$T_{RxN} = \begin{bmatrix}t_{11} & t_{12} & t_{13} & \ldots & t_{1N} \\t_{21} & t_{22} & t_{23} & \; & t_{2N} \\\; & \vdots & \; & \ddots & \vdots \\t_{R\; 1} & t_{R\; 2} & t_{R\; 3} & \ldots & t_{RN}\end{bmatrix}$

where R rows of the transform are R bases of the N dimensional space.The inverse transform matrix for RT is the transpose of its forwardtransform.

FIG. 15A shows a process (1501) of a reduced forward transform and aprocess (1502) of a reduced inverse transform. T represents an RSTtransform matrix having a dimension of R×N, and T^(T) represents atranspose matrix of T and T^(T) has a dimension of N×R.

In RST8×8, a reduction factor of 4 (¼ size) can be realized. Forexample, instead of 64×64, which is a conventional 8×8 non-separabletransform matrix size, a 16×64 direct matrix can be used. The 64×16inverse RST matrix can be used at the decoder side to generate core(primary) transform coefficients in an 8×8 top-left region. The forwardRST8×8 uses 16×64 (or 8×64 for 8×8 block) matrices so that the forwardRST8×8 produces non-zero coefficients only in the top-left 4×4 regionwithin the given 8×8 top-left region. In other words, when RST isapplied, a region in the 8×8 top-left region that is outside thetop-left 4×4 region has only zero coefficients. For RST4×4, 16×16 (or8×16 for 4×4 block) direct matrix multiplication can be applied.

In addition, for RST8×8, to further reduce the transform matrix size,instead of the using the whole top-left 8×8 coefficients (64coefficients in shaded sub-blocks in FIG. 15B) of a residual block(1510) as input for calculating a secondary transform, the top-leftthree 4×4 sub-block coefficients (48 coefficients in shaded sub-blocksin FIG. 15C) of the residual block (1510) are used as the input forcalculating the secondary transform. Accordingly, a 16×64 transformmatrix is used in FIG. 15B, and a 16×48 transform matrix is used in FIG.15C.

In an example, an inverse RST is conditionally applied where the inverseRST is applied when the following two conditions are satisfied: (i) ablock size (e.g., a width W and/or a height H of the block) is greaterthan or equal to a threshold (e.g., W>=4 and H>=4), and (ii) a transformskip mode flag is equal to zero. For example, if both width (W) andheight (H) of a transform coefficient block is greater than 4, then theRST8×8 is applied to the top-left 8×8 region of the transformcoefficient block. Otherwise, the RST4×4 is applied on the top-leftmin(8, W×min(8, H) region of the transform coefficient block.

In an example, when an RST index is equal to 0,RST is not applied.Otherwise, RST is applied, and a kernel is chosen with the RST index. Inan example, RST is applied for an intra CU (e.g., a CU coded in intraprediction or intra mode) in both intra and inter slices, and for bothluma and chroma. If a dual tree is enabled, RST indices for luma andchroma are signaled separately. For inter slice (the dual tree isdisabled), a single RST index is signaled and used for both luma andchroma. When an ISP mode is selected, RST is disabled, and RST index isnot signaled.

In an example, an RST matrix can be selected from four transform sets,each of which consists of two transforms. Which transform set is appliedcan be determined based on an intra prediction mode as follows. When oneof three cross component linear model (CCLM) modes is indicated, atransform set 0 can be selected. Otherwise, the transform set selectioncan be performed according to a table (1550) shown in FIG. 15D. An index(e.g., an IntraPredMode) to access the Table (1550) can be in a range of[−14, 80], which is a transformed mode index used for a wide angle intraprediction for example. An example of the intra prediction modes isshown in FIG. 16B. In an example, an index to access the Table (1550)can be in a range of [−14, 83] or any suitable range.

FIG. 16A shows an illustration of exemplary intra prediction directionsand the intra prediction modes used in HEVC. In HEVC, there are total 35intra prediction modes (mode 0 to mode 34). The mode 0 and mode 1 arenon-directional modes, among which mode 0 is planar mode (labeled asIntra Planar in FIG. 16A) and mode 1 is DC mode (labeled as Intra DC inFIG. 16A). The modes 2-34 are directional modes, among which mode 10 isa horizontal mode, mode 26 is a vertical mode, and mode 2, mode 18 andmode 34 are diagonal modes. In some examples, the intra prediction modesare signaled by three most probable modes (MPMs) and 32 remaining modes.

FIG. 16B shows an illustration of exemplary intra prediction directionsand intra prediction modes in some examples (e.g., VVC). There are total95 intra prediction modes (mode −14 to mode 80), among which mode 18 isa horizontal mode, mode 50 is a vertical mode, and mode 2, mode 34 andmode 66 are diagonal modes. Modes −1˜−14 and Modes 67˜80 are calledwide-angle intra prediction (WAIP) modes.

Multi-line intra prediction can use more reference lines for intraprediction. A reference line can include multiple samples in a picture.In an example, the reference line includes samples in a row and samplesin a column. In an example, an encoder can determine and signal areference line used to generate the intra predictor. An index (alsoreferred to as a reference line index) indicating the reference line canbe signaled before intra prediction mode(s). In an example, only theMPMs are allowed when a nonzero reference line index is signaled. FIG.17 shows an example of 4 reference lines for a coding block (1710). Inthe example shown in FIG. 17, a reference line can include six segments,i.e., Segment A to F. The reference line 3 can include a top-leftreference sample. The Segment A and F can be padded with closest samplesfrom the Segment B and E, respectively. In some examples, such as inHEVC, only one reference line (e.g., the reference line 0 that isadjacent to the coding block (1710)) is used for intra prediction. Insome examples, such as in VVC, multiple reference lines (e.g., thereference lines 0, 1, and 3) are used for intra prediction.

Intra sub-partition (ISP) coding mode can be used. In the ISP codingmode, a luma intra-predicted block can be divided vertically orhorizontally into 2 or 4 sub-partitions depending on a block size.

FIG. 18 shows a Table 4 that associates a number of sub-partitions witha block size. For example, when the block size is 4×4, no partition isperformed on the block in the ISP coding mode. When the block size is4×8 or 8×4, then the block is partitioned into two sub-partitions in theISP coding mode. For all other block sizes that are larger than 4×8 or8×4, the block is partitioned into four sub-partitions. FIG. 19 shows anexample of sub-partitions of a block having a size 4×8 or 8×4. FIG. 20shows another example of sub-partitions of a block having a size otherthan 4×8, 8×4 and 4×4, for example, the block size is larger than 4×8and 8×4. In an example, all of the sub-partitions satisfy a condition ofhaving at least 16 samples. For chroma components, ISP is not applied.

In some examples, for each of the sub-partitions, the decoder canentropy decode coefficients that are send from the encoder to thedecoder, then the decoder inverse quantizes and inverse transforms thecoefficients to generate residuals (or residual data) for thesub-partition. Further, when the sub-partition is intra predicted by thedecoder, the decoder can add the residuals with the intra predictionresults to obtain the reconstructed samples of the sub-partition.Therefore, the reconstructed samples of each sub-partition can beavailable to generate the prediction of next sub-partition(s) to bereconstructed. The process described above can be repeated for the nextsub-partition(s) and so on. All sub-partitions share the same intraprediction mode in an example. In some examples, in ISP, eachsub-partition can be regarded as a TU, since the transform andreconstruction is performed individually for each sub-partition.

In some examples, the ISP algorithm is only tested with intra predictionmodes that are part of the MPM list. For this reason, when a block usesISP, then the MPM flag can be inferred to be one. Besides, when ISP isused for a certain block, then the MPM list can be modified to excludethe DC mode and to prioritize horizontal intra prediction modes for theISP horizontal partition (or horizontal split) and vertical intraprediction modes for the vertical partition (or vertical split) in someexamples.

FIGS. 21A-21D show examples of different YUV formats or chroma formats.Each chroma format may define a different down-sampling grid ofdifferent color components.

A secondary transform can refer to a NSST, a RST (or RT), or the like. Asecondary transform index can refer to a NSST index, a RST index, or thelike. In an example, the second transform index indicates a secondtransform (also referred to as a second transform candidate). The secondtransform index can be signaled at a CU level. For example, a NSST indexor a RST index is signaled at a CU-level for a CU. Whether to signal thesecondary transform index can depend on a number of non-zerocoefficients of the CU. Thus, a decoder may loop all TUs included in theCU to determine the number of non-zero coefficients of the CU. In someembodiments, the process is relatively complicated.

In certain secondary transform (e.g., RST) designs, when a singlesplitting tree is used among different color components in a CUincluding, for example, a luma component and two chroma components, anumber of non-zero coefficients in the CU can be counted to determinewhether a secondary transform index is signaled or not. However, for a4×N or N×4 luma block, a corresponding chroma block is 2×N/2 or N/2×2 ina YUV4:2:0 format. Thus, a secondary transform such as RST is notapplied for the chroma block and a number of non-zero coefficients ofthe chroma block does not need to be counted.

In some examples, a second transform such as NSST or RST is not enabledfor ISP. This can limit the full benefit of secondary transforms interms of coding efficiency.

Embodiments described herein may be used separately or combined in anyorder. Further, the embodiments may be implemented by processingcircuitry (e.g., one or more processors or one or more integratedcircuits) in an encoder, a decoder, or the like. In one example, the oneor more processors can execute a program that is stored in anon-transitory computer-readable medium. In some examples, a block maybe a prediction block, a coding block, a CU, or the like.

In the disclosure, embodiments for DST-7 of an MTS candidate may beapplicable to DST-4, and embodiments for DCT-8 of an MTS candidate maybe applicable to DCT-4. Further, references to NSST may also apply toRST, which is an example of an alternative design of a non-separablesecondary transform, in some embodiments.

A high-level syntax (HLS) element can refer to a Video Parameter Set(VPS), a Sequence Parameter Set (SPS), a Picture Parameter Set (PPS), aSlice header, a Tile header, a Tile group header, or the like. A CTUheader can refer to syntax element(s) signaled for a CTU, e.g., asheader information. In an example, a CTU size is a maximum CU size. A TUsize may refer to a maximum width and/or a height, or an area of a TU.

In general, when a luma size (represented by luma samples) of a certainunit (e.g., a TU, a CU) is known, a corresponding chroma size that isspecified by a number of chroma samples can be obtained. In an example,a YUV format of 4:2:0 is used and a CU has a CU size of 64×64 lumasamples (or 64×64-L). Accordingly, the CU has a CU size of 32×32 chromasamples (or 32×32-C). The CU size can be referred to as 64×64-L,32×32-C, or 64×64-L/32×32-C. The CU can include a luma block and twochroma blocks where the luma block has 64×64 luma samples and each ofthe two chroma blocks has 32×32 chroma samples. The description can beadapted for a TU. For purposes of brevity, the descriptions are omitted.

A TU size can be represented using luma samples in the TU. For example,a maximum TU size of M samples refers to a maximum TU size of M lumasamples. Similarly, a CU size can be represented using luma samples in aCU. The TU size and the CU size can be represented using chroma samplesor a combination of luma and chroma samples in other embodiments.

A unit size may refer to a width, a height, and/or an area of the unit.For example, a maximum TU size may refer to a width, a height, and/or anarea of a maximum TU. In general, a TU, a CU, or the like can have anysuitable shape, including a rectangular shape, a square shape, an ‘L’shape, or any suitable shape. When the shape of the unit is irregular,such as an ‘L’ shape, the unit size can specify an area of the unit.

In some embodiments, a maximum TU size (also referred to as a maximumsize of a TU) can be signaled in a coded video bitstream, such as in HLS(e.g., a SPS and a PPS). The maximum TU size can be signaled in terms ofluma samples or chroma samples.

In some embodiments, a maximum TU size can be stored in an encoderand/or a decoder, thus the maximum TU size is not signaled. In oneexample, the maximum TU size can be stored in profile and/or leveldefinitions. The maximum TU size can be stored in terms of luma orchroma samples.

According to aspects of the disclosure, whether a secondary transform isallowed for a CU can be determined based on a size of the CU (or a CUsize). In an example, whether a secondary transform is allowed for theCU can be determined based on the CU size and a CU size threshold. Whenthe CU size is less than or equal to the CU size threshold, thesecondary transform is determined to be allowed, and when the CU size islarger than the CU size threshold, the secondary transform is determinednot to be allowed. In an example, when the secondary transform isdetermined not to be allowed, a secondary transform index is notsignaled. Thus, when a decoder determines that the second transform isnot allowed, the decoder can also determine that the second transformindex is not signaled. In an example, the CU size may refer to a widthand/or a height of the CU, such as 64 samples. In an example, the CUsize may refer to an area of the CU, such as 64×64 samples.

In an example, the CU size threshold (e.g., CU width threshold, CUheight threshold, or CU area threshold) is limited to be not larger thana maximum size of a TU in the CU. The maximum size of the TU can besignaled in the HLS. The maximum size of the TU can also be pre-definedand stored in a decoder.

As described above, in some examples, a decoder may loop TUs in a CU todetermine a number of non-zero coefficients of the CU and then determinewhether a secondary transform index is signaled. According to aspects ofthe disclosure, instead of counting the number of non-zero coefficientsof the CU, whether the secondary transform index is signaled can bedetermined based on a last position of non-zero transform coefficients(or a last non-zero coefficient position) of a first CB of the CU. Thefirst CB can be any suitable block, such as a luma block, a chromablock, or the like, that is in the CU. The second transform index canindicate a selected second transform for a second CB in the CU.

According to aspects of the disclosure, whether to perform the secondarytransform on the second CB in the CU can be determined based on whetherthe secondary transform index is determined to be signaled. Further,when the secondary transform is determined to be performed, a sample inthe second CB can be reconstructed after the secondary transformindicated by the second transform index is performed on the second CB.Alternatively, when the secondary transform is determined not to beperformed, the sample in the second CB can be reconstructed withoutperforming the secondary transform on the second CB. The second CB canbe any suitable block, such as a luma block, a chroma block, or thelike, that is in the CU. In an example, the first CB is the second CB.In another example, the first CB is different from the second CB.

In an example, the CU includes a luma block. The first CB is the lumablock. The last position is a last non-zero luma coefficient positionfor the luma block. Accordingly, whether the second transform index issignaled is determined based on the last luma position. The second CB isalso the luma block or the first CB.

In some embodiments, additional information can be included to determinewhether the secondary transform index is signaled, as described below.

In an example, the CU includes a luma block and a chroma block. Thefirst CB is the luma block. The last position is a last non-zero lumacoefficient position for the luma block. The additional information caninclude a last non-zero chroma coefficient position for the chromablock. Accordingly, whether the second transform index is signaled isdetermined based on the last non-zero luma coefficient position and thelast non-zero chroma coefficient position for the chroma block. Thesecond CB can be one of: the luma block and the chroma block.

In an example, the CU includes a luma block and two chroma blocks (e.g.,a chroma block I and a chroma block II). The first CB is the luma block.The last position is a last non-zero luma coefficient position for theluma block. The additional information can include a last non-zerochroma coefficient position I for the chroma block I and a last non-zerochroma coefficient position II for the chroma block II. Accordingly,whether the second transform index is signaled can be determined basedon the last non-zero luma coefficient position, the last non-zero chromacoefficient position I of non-zero transform coefficients for the chromablock I, and the last non-zero chroma coefficient position II ofnon-zero transform coefficients for the chroma block II. The second CBcan be one of: the luma block, the chroma block I, and the chroma blockII.

As described above, whether the secondary transform index is signaledcan be determined based on the last non-zero coefficient position of thefirst CB of the CU. The last non-zero coefficient position can include ahorizontal component (e.g., last_pos_x) and a vertical component (e.g.,last_pos_y), and thus whether the secondary transform index is signaledcan be determined based on the horizontal component and/or the verticalcomponent. The horizontal component and the vertical component can be aninteger that is 0 or larger than 0. The vertical component can be aninteger that is 0 or larger than 0.

In an embodiment, the horizontal component can be compared with a firstthreshold and/or the vertical component can be compared with a secondthreshold. The first threshold can be identical to the second threshold.Alternatively, the first threshold can be different from the secondthreshold. The first threshold and/or the second threshold can be apositive integer, such as 1, 2, 3, or the like.

In an example, whether the horizontal component is less than the firstthreshold and whether the vertical component is less than the secondthreshold can be determined. When the horizontal component is determinedto be less than the first threshold and the vertical component isdetermined to be less than the second threshold, the secondary transformindex can be determined not to be signaled.

In an example, whether the horizontal component is larger than or equalto the first threshold can be determined. Further, whether the verticalcomponent is larger than or equal to the second threshold can bedetermined. When the horizontal component is determined to be largerthan or equal to the first threshold and the vertical component isdetermined to be larger than or equal to the second threshold, thesecondary transform index can be determined to be signaled.

In an embodiment, whether a sum of the horizontal component and thevertical component of the last position is less than a third thresholdcan be determined. When the sum is determined to be less than the thirdthreshold, the secondary transform index can be determined not to besignaled. The third threshold can be a positive integer, such as 1, 2,3, or the like. When the sum is determined to be larger than or equal tothe third threshold, the secondary transform index can be determined tobe signaled.

In an embodiment, whether a minimum one of the horizontal component andthe vertical component is less than a fourth threshold can bedetermined. When the minimum one of the horizontal component and thevertical component is determined to be less than the fourth threshold,the secondary transform index is determined not to be signaled. Thefourth threshold can be a positive integer, such as 1, 2, 3, or thelike.

In an embodiment, whether a maximum one of the horizontal component andthe vertical component is less than a fifth threshold can be determined.When the maximum one is determined to be less than the fifth threshold,the secondary transform index is determined not to be signaled. Thefifth threshold can be a positive integer, such as 1, 2, 3, or the like.The fourth threshold can be identical to the fifth threshold.Alternatively, the fourth threshold can be different from the fifththreshold.

In an embodiment, the first CB is a luma block in the CU. Further, theCU includes a chroma block. However, whether the secondary transformindex is signaled is determined based on only the last non-zero lumacoefficient position of the luma block of the CU. Therefore, a lastnon-zero chroma coefficient position for the chroma block is notconsidered in determining whether the secondary transform index issignaled.

In an embodiment, the CU includes the luma block and the chroma block asdescribed above. Whether the second transform index is signaled isdetermined based on the last non-zero luma coefficient position for theluma block and the last non-zero chroma coefficient position for thechroma block. Similarly, the last non-zero luma coefficient position caninclude a luma horizontal component and a luma vertical component, andthe last non-zero chroma coefficient position can include a chromahorizontal component and a chroma vertical component. Thus, whether thesecondary transform index is signaled can be determined based on theluma horizontal component, the luma vertical component, the chromahorizontal component, and/or the chroma vertical component.

In an embodiment, one or more of the respective horizontal component andthe vertical component of each of the last non-zero luma and chromacoefficient positions can be compared with a respective threshold, suchas 1, 2, 3, or the like. In an example, whether the one or more of therespective horizontal component and the vertical component of each ofthe last non-zero luma and chroma coefficient positions is less than therespective threshold can be determined. When the one or more of therespective horizontal component and the vertical component of each ofthe last non-zero luma and chroma coefficient positions is less than therespective threshold, the secondary transform index can be determinednot to be signaled.

In an embodiment, a horizontal sum is obtained by summing the lumahorizontal component and the chroma horizontal component, and a verticalsum is obtained by summing the luma vertical component and the chromavertical component. Whether each of the horizontal sum and the verticalsum is less than a respective threshold can be determined. When each ofthe horizontal sum and the vertical sum is less than the respectivethreshold, the secondary transform index can be determined not to besignaled.

In an example, a first sum of the luma horizontal component and the lumavertical component is determined, and a second sum of the chromahorizontal component and the chroma vertical component is determined.Whether each of the first sum and the second sum is less than arespective threshold can be determined. When each of the first sum andthe second sum is determined to be less than the respective threshold,the secondary transform index can be determined not to be signaled.

In an embodiment, a total sum is obtained by summing the first sum andthe second sum, when the total sum is determined to be less than athreshold, the secondary transform index can be determined not to besignaled.

In an embodiment, a first minimum of the luma horizontal component andthe luma vertical component is determined, and a second minimum of thechroma horizontal component and the chroma vertical component isdetermined. Whether each of the first minimum and the second minimum isless than a respective threshold can be determined. When each of thefirst minimum and the second minimum is determined to be less than therespective threshold, the secondary transform index can be determinednot to be signaled. The above description can be adapted for using afirst maximum of the luma horizontal component and the luma verticalcomponent and a second maximum of the chroma horizontal component andthe chroma vertical component to determine whether the second transformindex is signaled.

Similarly, a minimum of the horizontal sum and the vertical sum can beused to determine whether the second transform index is signaled or not.A maximum of the horizontal sum and the vertical sum can be used todetermine whether the second transform index is signaled or not.

According to aspects of the disclosure, when determining a number ofnon-zero transform coefficients for a CU, non-zero transformcoefficients in a CB of the CU can be counted when a size of the CB(also referred to as a CB size) is larger than or equal to a sizethreshold, such as 4. In an example, when the CB size is less than thesize threshold, the non-zero transform coefficients in the CB are notcounted, i.e., are not included in the number of non-zero transformcoefficients for the CU. The size threshold can be pre-defined andstored in a decoder. The size threshold can be signaled explicitly, forexample, from an encoder to a decoder. Further, when the number ofnon-zero transform coefficients in the CU is less than a numberthreshold, a secondary transform index can be determined not to besignaled.

In an example, a color format for the CU is YUV 4:2:0. The CU includes aluma block and two chroma blocks that are co-located with the lumablock. The size threshold is 4. When the luma block has a size of 4×N orN×4 where N can refer to a width or a height of the luma block and canbe larger than or equal to 4, the two chroma blocks have a size of 2×N/2or N/2×2. In an example, N is a positive even number. The number ofnon-zero transform coefficients for the CU is determined from only theluma block without considering the two chroma blocks. A width or aheight of each of the two chroma blocks is less than the size threshold.In another example, the size threshold is 4×4, and the CB size of 2×N/2or N/2×2 is also less than the size threshold where N can be a positiveeven number. A secondary transform is not performed on the two chromablocks.

In an example, a color format for the CU is YUV 4:2:2. The CU includes aluma block and two chroma blocks that are co-located with the lumablock. The size threshold is 4. When the luma block has a size of 4×Nwhere N is larger than or equal to 4, the two chroma blocks have a sizeof 2×N. The number of non-zero transform coefficients for the CU isdetermined from only the luma block without considering the two chromablocks. A width (e.g., 2) of each of the two chroma blocks is less thanthe size threshold. A secondary transform is not performed on the twochroma blocks.

A secondary transform can be performed on a first coefficient block(such as a TB) to obtain a 4×2 (or 2×4) first sub-block that includes atleast one non-zero coefficient. For example, a RST is applied on a 4×4first TB (e.g., the first coefficient block) to obtain a second TB thatincludes a first sub-block and a second sub-block. The first sub-blockof 4×2 (or 2×4) includes at least one non-zero coefficient. Coefficientsin the second sub-block of 4×2 (or 2×4) are regarded as 0. Thus, a 4×2(or 2×4) sub-block scanning of the first sub-block (i.e., the 4×2 (or2×4) coefficient scanning order) is applied for entropy coding thesecond TB. In an example, the 4×2 (or 2×4) coefficient scanning order isa same scanning order that is applied for entropy coding of a 4×2 (or2×4) chroma block.

Similarly, a secondary transform can be performed on a first coefficientblock (such as a TB) to obtain an 8×4 (or 4×8) first sub-block thatincludes at least one non-zero coefficient, for example, when the firstcoefficient block is larger than 8×4 (or 4×8). For example, a RST isapplied on a 8×8 first TB (e.g., the first coefficient block) to obtaina second TB that includes a first sub-block and a second sub-block. Thefirst sub-block of 8×4 (or 4×8) includes at least one non-zerocoefficient. Coefficients in the second sub-block of 8×4 (or 4×8) areregarded as 0. Thus, an 8×4 (or 4×8) block scanning of the firstsub-block (i.e., the 8×4 (or 4×8) coefficient scanning order) is appliedfor entropy coding the second TB. In an example, the 8×4 (or 4×8)coefficient scanning order is a same scanning order that is applied forentropy coding of a 8×4 (or 4×8) chroma block.

FIG. 22 shows a flow chart outlining a process (2200) according to anembodiment of the disclosure. The process (2200) can be used in thereconstruction of a block coded in intra mode, so to generate aprediction block for the block under reconstruction. In some examples,the process (2200) can be used in the reconstruction of a block coded ininter mode. In various embodiments, the process (2200) are executed byprocessing circuitry, such as the processing circuitry in the terminaldevices (310), (320), (330) and (340), the processing circuitry thatperforms functions of the video encoder (403), the processing circuitrythat performs functions of the video decoder (410), the processingcircuitry that performs functions of the video decoder (510), theprocessing circuitry that performs functions of the video encoder (603),and the like. In some embodiments, the process (2200) is implemented insoftware instructions, thus when the processing circuitry executes thesoftware instructions, the processing circuitry performs the process(2200). The process starts at (S2201) and proceeds to (S2210).

At (S2210), coded information of a CU can be decoded from a coded videobitstream. The coded information can indicate a last position ofnon-zero transform coefficients (or a last non-zero coefficientposition) of a first CB of the CU. In an example, the CU can include aluma block and a chroma block. The first CB can be the luma block or thechroma block.

At (S2220), whether a secondary transform index is signaled in the codedinformation can be determined based on the last non-zero coefficientposition, as described above. The second transform index can indicate asecond transform to be performed on a second CB in the CU. The second CBcan be the luma block or the chroma block.

In an example, the last non-zero coefficient position can include ahorizontal component and a vertical component, and whether the secondarytransform index is signaled in the coded information can be determinedbased on the horizontal component and/or the vertical component. Asdescribed above, additional information can be used to determine whetherthe secondary transform index is signaled and step (S2220) can besuitably adapted to include the additional information. In an example,when the secondary transform index is determined to be signaled, theprocess (2200) proceeds to (S2230). Otherwise, the process (2200)proceeds to (S2250)

At (S2230), whether to perform the secondary transform on the second CBcan be determined based on whether the secondary transform index isdetermined to be signaled in the coded information. In some examples,when the secondary transform index is determined to be signaled, thesecondary transform is determined to be performed. When the secondarytransform is determined to be performed, the process (2200) proceeds to(S2240). Otherwise, the process (2200) proceeds to (S2250).

At (S2240), the secondary transform indicated by the second transformindex is performed on the second CB. The secondary transform can beNSST. The secondary transform can be RST including a zero-out method.For example, when the second CB is 8×8, RST is applied on the second CBto obtain a transformed block that includes a first sub-block of 8×4 anda second sub-block of 8×4. The first sub-block includes at least onenon-zero coefficient. Coefficients in the second sub-block are notcalculated and regarded as 0.

At (S2250), a sample in the second CB can be reconstructed. The process(2200) then proceeds to (S2299) and terminates.

The process (2200) can be suitably adapted as described above. Forexample, one or more steps can be modified, omitted, or combined. In anexample, steps (S2220) and (S2230) are combined. Additional step(s) canalso be added. An order that the process (2200) is executed can also bemodified.

FIG. 23 shows a flow chart outlining a process (2300) according to anembodiment of the disclosure. The process (2300) can be used in thereconstruction of a block coded in intra mode, so to generate aprediction block for the block under reconstruction. In some examples,the process (2300) can be used in the reconstruction of a block coded ininter mode. In various embodiments, the process (2300) are executed byprocessing circuitry, such as the processing circuitry in the terminaldevices (310), (320), (330) and (340), the processing circuitry thatperforms functions of the video encoder (403), the processing circuitrythat performs functions of the video decoder (410), the processingcircuitry that performs functions of the video decoder (510), theprocessing circuitry that performs functions of the video encoder (603),and the like. In some embodiments, the process (2300) is implemented insoftware instructions, thus when the processing circuitry executes thesoftware instructions, the processing circuitry performs the process(2300). The process starts at (S2301) and proceeds to (S2310).

At (S2310), coding information of a CU can be decoded from a coded videobitstream where the coding information indicates a size of the CU.

At (S2320), whether a secondary transform is allowed can be determinedbased on the size of the CU and a CU size threshold. When the size ofthe CU is less than or equal to the CU size threshold, the secondarytransform is determined to be allowed. The process (2300) proceeds to(S2330). When the size of the CU is larger than the CU size threshold,the secondary transform is determined not to be allowed and the process(2300) proceeds to (S2350).

At (S2330), whether to perform a secondary transform on a CB in the CUcan be determined, for example, based on whether the secondary transformindex is signaled, as described above. When the secondary transform isdetermined to be performed, the process (2300) proceeds to (S2340).Otherwise, the process (2300) proceeds to (S2350).

At (S2340), the secondary transform indicated by the second transformindex is performed on the CB, similar to step (S2240).

At (S2350), a sample in the CB can be reconstructed. The process (2300)then proceeds to (S2399) and terminates.

The process (2300) can be suitably adapted. For example, one or moresteps can be modified. Additional step(s) can also be added.

The process (2200) and process (2300) can be suitably combined. Forexample, (S2310) and (S2320) can be implemented, followed by(S2210)-(S2250).

The techniques described above, can be implemented as computer softwareusing computer-readable instructions and physically stored in one ormore computer-readable media. For example, FIG. 24 shows a computersystem (2400) suitable for implementing certain embodiments of thedisclosed subject matter.

The computer software can be coded using any suitable machine code orcomputer language, that may be subject to assembly, compilation,linking, or like mechanisms to create code comprising instructions thatcan be executed directly, or through interpretation, micro-codeexecution, and the like, by one or more computer central processingunits (CPUs), Graphics Processing Units (GPUs), and the like.

The instructions can be executed on various types of computers orcomponents thereof, including, for example, personal computers, tabletcomputers, servers, smartphones, gaming devices, internet of thingsdevices, and the like.

The components shown in FIG. 24 for computer system (2400) are exemplaryin nature and are not intended to suggest any limitation as to the scopeof use or functionality of the computer software implementingembodiments of the present disclosure. Neither should the configurationof components be interpreted as having any dependency or requirementrelating to any one or combination of components illustrated in theexemplary embodiment of a computer system (2400).

Computer system (2400) may include certain human interface inputdevices. Such a human interface input device may be responsive to inputby one or more human users through, for example, tactile input (such as:keystrokes, swipes, data glove movements), audio input (such as: voice,clapping), visual input (such as: gestures), olfactory input (notdepicted). The human interface devices can also be used to capturecertain media not necessarily directly related to conscious input by ahuman, such as audio (such as: speech, music, ambient sound), images(such as: scanned images, photographic images obtain from a still imagecamera), video (such as two-dimensional video, three-dimensional videoincluding stereoscopic video).

Input human interface devices may include one or more of (only one ofeach depicted): keyboard (2401), mouse (2402), trackpad (2403), touchscreen (2410), data-glove (not shown), joystick (2405), microphone(2406), scanner (2407), camera (2408).

Computer system (2400) may also include certain human interface outputdevices. Such human interface output devices may be stimulating thesenses of one or more human users through, for example, tactile output,sound, light, and smell/taste. Such human interface output devices mayinclude tactile output devices (for example tactile feedback by thetouch-screen (2410), data-glove (not shown), or joystick (2405), butthere can also be tactile feedback devices that do not serve as inputdevices), audio output devices (such as: speakers (2409), headphones(not depicted)), visual output devices (such as screens (2410) toinclude CRT screens, LCD screens, plasma screens, OLED screens, eachwith or without touch-screen input capability, each with or withouttactile feedback capability—some of which may be capable to output twodimensional visual output or more than three dimensional output throughmeans such as stereographic output; virtual-reality glasses (notdepicted), holographic displays and smoke tanks (not depicted)), andprinters (not depicted).

Computer system (2400) can also include human accessible storage devicesand their associated media such as optical media including CD/DVD ROM/RW(2420) with CD/DVD or the like media (2421), thumb-drive (2422),removable hard drive or solid state drive (2423), legacy magnetic mediasuch as tape and floppy disc (not depicted), specialized ROM/ASIC/PLDbased devices such as security dongles (not depicted), and the like.

Those skilled in the art should also understand that term “computerreadable media” as used in connection with the presently disclosedsubject matter does not encompass transmission media, carrier waves, orother transitory signals.

Computer system (2400) can also include an interface to one or morecommunication networks. Networks can for example be wireless, wireline,optical. Networks can further be local, wide-area, metropolitan,vehicular and industrial, real-time, delay-tolerant, and so on. Examplesof networks include local area networks such as Ethernet, wireless LANs,cellular networks to include GSM, 3G, 4G, 5G, LTE and the like, TVwireline or wireless wide area digital networks to include cable TV,satellite TV, and terrestrial broadcast TV, vehicular and industrial toinclude CANBus, and so forth. Certain networks commonly require externalnetwork interface adapters that attached to certain general purpose dataports or peripheral buses (2449) (such as, for example USB ports of thecomputer system (2400)); others are commonly integrated into the core ofthe computer system (2400) by attachment to a system bus as describedbelow (for example Ethernet interface into a PC computer system orcellular network interface into a smartphone computer system). Using anyof these networks, computer system (2400) can communicate with otherentities. Such communication can be uni-directional, receive only (forexample, broadcast TV), uni-directional send-only (for example CANbus tocertain CANbus devices), or bi-directional, for example to othercomputer systems using local or wide area digital networks. Certainprotocols and protocol stacks can be used on each of those networks andnetwork interfaces as described above.

Aforementioned human interface devices, human-accessible storagedevices, and network interfaces can be attached to a core (2440) of thecomputer system (2400).

The core (2440) can include one or more Central Processing Units (CPU)(2441), Graphics Processing Units (GPU) (2442), specialized programmableprocessing units in the form of Field Programmable Gate Areas (FPGA)(2443), hardware accelerators for certain tasks (2444), and so forth.These devices, along with Read-only memory (ROM) (2445), Random-accessmemory (2446), internal mass storage such as internal non-useraccessible hard drives, SSDs, and the like (2447), may be connectedthrough a system bus (2448). In some computer systems, the system bus(2448) can be accessible in the form of one or more physical plugs toenable extensions by additional CPUs, GPU, and the like. The peripheraldevices can be attached either directly to the core's system bus (2448),or through a peripheral bus (2449). Architectures for a peripheral businclude PCI, USB, and the like.

CPUs (2441), GPUs (2442), FPGAs (2443), and accelerators (2444) canexecute certain instructions that, in combination, can make up theaforementioned computer code. That computer code can be stored in ROM(2445) or RAM (2446). Transitional data can be also be stored in RAM(2446), whereas permanent data can be stored for example, in theinternal mass storage (2447). Fast storage and retrieve to any of thememory devices can be enabled through the use of cache memory, that canbe closely associated with one or more CPU (2441), GPU (2442), massstorage (2447), ROM (2445), RAM (2446), and the like.

The computer readable media can have computer code thereon forperforming various computer-implemented operations. The media andcomputer code can be those specially designed and constructed for thepurposes of the present disclosure, or they can be of the kind wellknown and available to those having skill in the computer software arts.

As an example and not by way of limitation, the computer system havingarchitecture (2400), and specifically the core (2440) can providefunctionality as a result of processor(s) (including CPUs, GPUs, FPGA,accelerators, and the like) executing software embodied in one or moretangible, computer-readable media. Such computer-readable media can bemedia associated with user-accessible mass storage as introduced above,as well as certain storage of the core (2440) that are of non-transitorynature, such as core-internal mass storage (2447) or ROM (2445). Thesoftware implementing various embodiments of the present disclosure canbe stored in such devices and executed by core (2440). Acomputer-readable medium can include one or more memory devices orchips, according to particular needs. The software can cause the core(2440) and specifically the processors therein (including CPU, GPU,FPGA, and the like) to execute particular processes or particular partsof particular processes described herein, including defining datastructures stored in RAM (2446) and modifying such data structuresaccording to the processes defined by the software. In addition or as analternative, the computer system can provide functionality as a resultof logic hardwired or otherwise embodied in a circuit (for example:accelerator (2444)), which can operate in place of or together withsoftware to execute particular processes or particular parts ofparticular processes described herein. Reference to software canencompass logic, and vice versa, where appropriate. Reference to acomputer-readable media can encompass a circuit (such as an integratedcircuit (IC)) storing software for execution, a circuit embodying logicfor execution, or both, where appropriate. The present disclosureencompasses any suitable combination of hardware and software.

Appendix A: Acronyms

JEM: joint exploration model

VVC: versatile video coding

BMS: benchmark set

MV: Motion Vector

HEVC: High Efficiency Video Coding

SEI: Supplementary Enhancement Information

VUI: Video Usability Information

GOPs: Groups of Pictures

TUs: Transform Units,

PUs: Prediction Units

CTUs: Coding Tree Units

CTBs: Coding Tree Blocks

PBs: Prediction Blocks

HRD: Hypothetical Reference Decoder

SNR: Signal Noise Ratio

CPUs: Central Processing Units

GPUs: Graphics Processing Units

CRT: Cathode Ray Tube

LCD: Liquid-Crystal Display

OLED: Organic Light-Emitting Diode

CD: Compact Disc

DVD: Digital Video Disc

ROM: Read-Only Memory

RAM: Random Access Memory

ASIC: Application-Specific Integrated Circuit

PLD: Programmable Logic Device

LAN: Local Area Network

GSM: Global System for Mobile communications

LTE: Long-Term Evolution

CANBus: Controller Area Network Bus

USB: Universal Serial Bus

PCI: Peripheral Component Interconnect

FPGA: Field Programmable Gate Areas

SSD: solid-state drive

IC: Integrated Circuit

CU: Coding Unit

While this disclosure has described several exemplary embodiments, thereare alterations, permutations, and various substitute equivalents, whichfall within the scope of the disclosure. It will thus be appreciatedthat those skilled in the art will be able to devise numerous systemsand methods which, although not explicitly shown or described herein,embody the principles of the disclosure and are thus within the spiritand scope thereof.

Appendix I

4×4 Transform

{64, 64, 64, 64} {83, 36, −36, −83} {64, −64, −64, 64} {36, −83, 83,−36}

8×8 Transform

{64, 64, 64, 64, 64, 64, 64, 64} {89, 75, 50, 18, −18, −50, −75, −89}{83, 36, −36, −83, −83, −36, 36, 83} {75, −18, −89, −50, 50, 89, 18,−75} {64, −64, −64, 64, 64, −64, −64, 64} {50, −89, 18, 75, −75, −18,89, −50} {36, −83, 83, −36, −36, 83, −83, 36} {18, −50, 75, −89, 89,−75, 50, −18}

16×16 Transform

{64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 } {90 87 80 70 57 43 259 −9−25−43−57−70−80−87−90} {89 75 50 18−18−50−75−89−89−75−50−18 18 50 7589} {87 57 9−43−80−90−70−25 25 70 90 80 43 −9−57−87} {83 36−36−83−83−3636 83 83 36−36−83−83−36 36 83} {80 9−70−87−25 57 90 43−43−90−57 25 87 70−9−80} {75−18−89−50 50 89 18−75−75 18 89 50−50−89−18 75} {70−43−87 9 9025−80−57 57 80−25−90 −9 87 43−70} {64−64−64 64 64−64−64 64 64−64−64 6464−64−64 64} {57−80−25 90 −9−87 43 70−70−43 87 9−90 25 80−57} {50−89 1875−75−18 89−50−50 89−18−75 75 18−89 50} { 43−90 57 25−87 70 9−80 80−9−70 87−25−57 90−43} { 36−83 83−36−36 83−83 36 36−83 83−36−36 83−83 36}{25−70 90−80 43 9−57 87−87 57 −9−43 80−90 70−25} {18−50 75−89 89−7550−18−18 50−75 89−89 75−50 18} { 9−25 43−57 70−80 87−90 90−87 80−7057−43 25 −9}

32×32 Transform

{64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 6464 64 64 64 64 64 64 64} {90 90 88 85 82 78 73 67 61 54 46 38 31 22 13 4−4−13−22−31−38−46−54−61−67−73−78−82−85−88−90−90} {90 87 80 70 57 43 25 9−9−25−43−57−70−80−87−90−90−87−80−70−57−43−25 −9 9 25 43 57 70 80 87 90}{90 82 67 46 22 −4−31−54−73−85−90−88−78−61−38−13 13 38 61 78 88 90 85 7354 31 4−22−46−67−82−90} {89 75 50 18−18−50−75−89−89−75−50−18 18 50 75 8989 75 50 18−18−50−75−89−89−75−50−18 18 50 75 89} {88 6731−13−54−82−90−78−46 −4 38 73 90 85 61 22−22−61−85−90−73−38 4 46 78 9082 54 13−31−67−88} {87 57 9−43−80−90−70−25 25 70 90 80 43 −9−57−87−87−57−9 43 80 90 70 25−25−70−90−80−43 9 57 87} {85 46−13−67−90−73−22 38 82 8854 −4−61−90−78−31 31 78 90 61 4−54−88−82−38 22 73 90 67 13−46−85} {83 36−36 −83−83−36 36 83 83 36−36−83−83−36 36 83 83 36−36−83−83−36 36 83 8336−36−83−83−36 36 83} {82 22−54−90−61 13 78 85 31−46−90−67 4 73 8838−38−88−73 −4 67 90 46−31−85−78−13 61 90 54−22−82} {80 9−70−87−25 57 9043−43−90−57 25 87 70 −9−80 −80 −9 70 87 25−57−90−43 43 90 57−25−87−70 980} {78 −4−82−73 13 85 67−22−88−61 31 90 54−38−90−46 46 90 38−54−90−3161 88 22−67−85−13 73 82 4−78} {75−18−89−50 50 89 18−75−75 18 8950−50−89−18 75 75−18−89−50 50 89 18−75−75 18 89 50−50−89−18 75}{73−31−90−22 78 67−38−90−13 82 61−46−88 −4 85 54−54−85 4 88 46−61−82 1390 38−67−78 22 90 31−73} {70−43−87 9 90 25−80 57 57 80−25−90 −9 8743−70−70 43 87 −9−90−25 80 57−57−80 25 90 9−87−43 70} {67−54−78 3885−22−90 4 90 13−88−31 82 46−73−61 61 73−46−82 31 88−13−90 −4 9022−85−38 78 54−67} {64−64−64 64 64−64−64 64 64−64−64 64 64−64−64 6464−64−64 64 64−64−64 64 64−64−64 64 64 −64 −64 64} {61−73−46 82 31−88−1390 −4−90 22 85−38−78 54 67−67−54 78 38 −85 −22 90 4−90 13 88−31−82 46 73−61} {57−80−25 90 −9−87 43 70−70−43 87 9−90 25 80−57−57 80 25−90 987−43−70 70 43−87 −9 90 −25−80 57} {54−85 −4 88−46−61 82 13−90 3867−78−22 90−31−73 73 31−90 22 78−67−38 90−13−82 61 46−88 4 85−54} {50−8918 75−75−18 89−50−50 89−18−75 75 18−89 50 50−89 18 75−75−18 89−50−5089−18−75 75 18−89 50} {46−90 38 54−90 31 61−88 22 67−85 13 73−82 4 78−78−4 82−73−13 85−67−22 88−61−31 9−54−38 90−46} {43−90 57 25−87 70 9−80 80−9−70 87−25−57 90−4−43 90−57−5 87−70 −9 80−80 9 70−87 25 57−90 43}{38−88 73 −4−67 90−46−31 85−78 13 61−90 54 22−82 82−22−54 90−61−13 78−8531 46−90 67 4−73 88−38} {36−83 83−36−36 83−83 36 36−83 83−36−36 83−83 3636−83 83−36−36 83−83 36 36−83 8−36−36 83−83 36} {31−78 90−61 4 54−8882−38−22 73−90 67−13−46 85−85 46 13−67 90−73 22 38−82 88−54 −4 61−9078−31} {25−70 90−80 43 9−57 87−87 57 −9−43 80−90 70−25−25 70−90 80−43 −957−87 87−57 9 43−80 90−70 25} {22−61 85−90 73−38 −4 46−78 90−82 54−13−3167−88 88−67 31 13−54 82−90 78−46 4 38−73 90−85 61−22} {18−50 75−89 89−7550−18−18 50−75 89−89 75−50 18 18−50 75−89 89−75 50−18−18 50−75 89−8975−50 18} {13−38 61−78 88−90 85−73 54−31 4 22−46 67−82 90−90 82−67 46−22−4 31−54 73−85 90−88 78−61 38−13} {43−57 70−80 87−90 90−87 80−70 57−4325 −9 −9 25−43 57−70 80−87 90−90 87−80 70−57 43−25 9} { 4−13 22−31 38−4654−61 67−73 78−82 85−88 90−90 90−90 88−85 82−78 73−67 61−54 46−38 31−2213 −4}

APPENDIX II 64-point DCT-2 core { {aa, aa, aa, aa, aa, aa, aa, aa, aa,aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,aa} {bf, bg, bh, bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, bv,bw, bx, by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck, -ck, -cj,-ci, -ch, -cg, -cf, -ce, -cd, -cc, -cb, -ca, -bz, -by, - bx, -bw, -bv,-bu, -bt, -bs, -br, -bq, -bp, -bo, -bn, -bm, -bl, -bk, -bj, -bi, -bh,-bg, -bf} {ap, aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd,be, -be, -bd, -bc, -bb, -ba, -az, - ay, -ax, -aw, -av, -au, -at, -as,-ar, -aq, -ap, -ap, -aq, -ar, -as, -at, -au, -av, -aw, -ax, -ay, -az,-ba, - bb, -bc, -bd, -be, be, bd, bc, bb, ba, az, ay, ax, aw, av, au,at, as, ar, aq, ap} {bg, bj, bm, bp, bs, bv, by, cb, ce, ch, ck, -ci,-cf, -cc, -bz, -bw, -bt, -bq, -bn, -bk, -bh, -bf, -bi, -bl, -bo, -br,-bu, -bx, -ca, -cd, -cg, -cj, cj, cg, cd, ca, bx, bu, br, bo, bl, bi,bf, bh, bk, bn, bq, bt, bw, bz, cc, cf, ci, -ck, -ch, -ce, -cb, -by,-bv, -bs, -bp, -bm, -bj, -bg} {ah, ai, aj, ak, al, am, an, ao, -ao, -an,-am, -al, -ak, -aj, -ai, -ah, -ah, -ai, -aj, -ak, -al, -am, - an, -ao,ao, an, am, al, ak, aj, ai, ah, ah, ai, aj, ak, al, am, an, ao, -ao,-an, -am, -al, -ak, - aj, -ai, -ah, -ah, -ai, -aj, -ak, -al, -am, -an,-ao, ao, an, am, al, ak, aj, ai, ah} {bh, bm, br, bw, cb, cg, -ck, -cf,-ca, -bv, -bq, -bl, -bg, -bi, -bn, -bs, -bx, -cc, -ch, cj, ce, bz, bu,bp, bk, bf, bj, bo, bt, by, cd, ci, -ci, -cd, -by, -bt, -bo, -bj, -bf,-bk, -bp, -bu, -bz, -ce, -cj, ch, cc, bx, bs, bn, bi, bg, bl, bq, bv,ca, cf, ck, -cg, -cb, -bw, -br, -bm, -bh} {aq, at, aw, az, bc, -be, -bb,-ay, -av, -as, -ap, -ar, -au, -ax, -ba, -bd, bd, ba, ax, au, ar, ap, as,av, ay, bb, be, -bc, -az, -aw, -at, -aq, -aq, -at, -aw, -az, -bc, be,bb, ay, av, as, ap, ar, au, ax, ba, bd, -bd, -ba, -ax, -au, -ar, -ap,-as, -av, -ay, -bb, -be, bc, az, aw, at, aq} {bi, bp, bw, cd, ck, -ce,-bx, -bq, -bj, -bh, -bo, -bv, -cc, -cj, cf, by, br, bk, bg, bn, bu, cb,ci, -cg, -bz, -bs, -bl, -bf, -bm, -bt, -ca, -ch, ch, ca, bt, bm, bf, bl,bs, bz, cg, -ci, -cb, -bu, -bn, -bg, -bk, -br, -by, -cf, cj, cc, bv, bo,bh, bj, bq, bx, ce, -ck, -cd, -bw, -bp, -bi} {ad, ae, af, ag, -ag, -af,-ae, -ad, -ad, -ae, -af, -ag, ag, af, ae, ad, ad, ae, af, ag, -ag,-af, - ae, -ad, -ad, -ae, -af, -ag, ag, af, ae, ad, ad, ae, af, ag, -ag,-af, -ae, -ad, -ad, -ae, -af, -ag, ag, af, ae, ad, ad, ae, af, ag, -ag,-af, -ae, -ad, -ad, -ae, -af, -ag, ag, af, ae, ad} {bj, bs, cb, ck, -cc,-bt, -bk, -bi, -br, -ca, -cj, cd, bu, bl, bh, bq, bz, ci, -ce, -bv, -bm,-bg, - bp, -by, -ch, cf, bw, bn, bf, bo, bx, cg, -cg, -bx, -bo, -bf,-bn, -bw, -cf, ch, by, bp, bg, bm, bv, ce, -ci, -bz, -bq, -bh, -bl, -bu,-cd, cj, ca, br, bi, bk, bt, cc, -ck, -cb, -bs, -bj} {ar, aw, bb, -bd,-ay, -at, -ap, -au, -az, -be, ba, av, aq, as, ax, bc, -bc, -ax, -as,-aq, -av, -ba, be, az, au, ap, at, ay, bd, -bb, -aw, -ar, -ar, -aw, -bb,bd, ay, at, ap, au, az, be, -ba, -av, -aq, -as, -ax, -bc, bc, ax, as,aq, av, ba, -be, -az, -au, -ap, -at, -ay, -bd, bb, aw, ar} {bk, bv, cg,-ce, -bt, -bi, -bm, -bx, -ci, cc, br, bg, bo, bz, ck, -ca, -bp, -bf,-bq, -cb, cj, by, bn, bh, bs, cd, -ch, -bw, -bl, -bj, -bu, -cf, cf, bu,bj, bl, bw, ch, -cd, -bs, -bh, -bn, -by, -cj, cb, bq, bf, bp, ca, -ck,-bz, -bo, -bg, -br, -cc, ci, bx, bm, bi, bt, ce, -cg, -bv, -bk} {ai, al,ao, -am, -aj, -ah, -ak, -an, an, ak, ah, aj, am, -ao, -al, -ai, -ai,-al, -ao, am, aj, ah, ak, an, -an, -ak, -ah, -aj, -am, ao, al, ai, ai,al, ao, -am, -aj, -ah, -ak, -an, an, ak, ah, aj, am, -ao, -al, -ai, -ai,-al, -ao, am, aj, ah, ak, an, -an, -ak, -ah, -aj, -am, ao, al, ai} {bl,by, -ck, -bx, -bk, -bm, -bz, cj, bw, bj, bn, ca, -ci, -bv, -bi, -bo,-cb, ch, bu, bh, bp, cc, -cg, -bt, -bg, -bq, -cd, cf, bs, bf, br, ce,-ce, -br, -bf, -bs, -cf, cd, bq, bg, bt, cg, -cc, -bp, -bh, -bu, -ch,cb, bo, bi, bv, ci, -ca, -bn, -bj, -bw, -cj, bz, bm, bk, bx, ck, -by,-bl} {as, az, -bd, -aw, -ap, -av, -bc, ba, at, ar, ay, -bc, -ax, -aq,-au, -bb, bb, au, aq, ax, be, -ay, -ar, -at, -ba, bc, av, ap, aw, bd,-az, -as, -as, -az, bd, aw, ap, av, be, -ba, -at, -ar, -ay, be, ax, aq,au, bb, -bb, -au, -aq, -ax, -be, ay, ar, at, ba, -be, -av, -ap, -aw,-bd, az, as} {bm, cb, -cf, -bq, -bi, -bx, cj, bu, bf, bt, ci, -by, -bj,-bp, -ce, cc, bn, bl, ca, -cg, -br, -bh, - bw, ck, bv, bg, bs, ch, -bz,-bk, -bo, -cd, cd, bo, bk, bz, -ch, -bs, -bg, -bv, -ck, bw, bh, br, cg,-ca, -bl, -bn, -cc, ce, bp, bj, by, -ci, -bt, -bf, -bu, -cj, bx, bi, bq,cf, -cb, -bm} {ab, ac, -ac, -ab, -ab, -ac, ac, ab, ab, ac, -ac, -ab,-ab, -ac, ac, ab, ab, ac, -ac, -ab, -ab, -ac, ac, ab, ab, ac, -ac, -ab,-ab, -ac, ac, ab, ab, ac, -ac, -ab, -ab, -ac, ac, ab, ab, ac, -ac, -ab,-ab, -ac, ac, ab, ab, ac, -ac, -ab, -ab, -ac, ac, ab, ab, ac, -ac, -ab,-ab, -ac, ac, ab} {bn, ce, -ca, -bj, -br, -ci, bw, bf, bv, -cj, -bs,-bi, -bz, cf, bo, bm, cd, -cb, -bk, -bq, -ch, bx, bg, bu, -ck, -bt, -bh,-by, cg, bp, bl, cc, -cc, -bl, -bp, -cg, by, bh, bt, ck, -bu, -bg, -bx,ch, bq, bk, cb, -cd, -bm, -bo, -cf, bz, bi, bs, cj, -bv, -bf, -bw, ci,br, bj, ca, -ce, -bn} {at, bc, -ay, -ap, -ax, bd, au, as, bb, -az, -aq,-aw, be, av, ar, ba, -ba, -ar, -av, -be, aw, aq, az, -bb, -as, -au, -bd,ax, ap, ay, -bc, -at, -at, -bc, ay, ap, ax, -bd, -au, -as, -bb, az, aq,aw, - be, -av, -ar, -ba, ba, ar, av, be, -aw, -aq, -az, bb, as, au, bd,-ax, -ap, -ay, bc, at} {bo, ch, -bv, -bh, -ca, cc, bj, bt, -cj, -bq,-bm, -cf, bx, bf, by, -ce, -bl, -br, -ck, bs, bk, cd, - bz, -bg, -bw,cg, bn, bp, ci, -bu, -bi, -cb, cb, bi, bu, -ci, -bp, -bn, -cg, bw, bg,bz, -cd, -bk, - bs, ck, br, bl, ce, -by, -bf, -bx, cf, bm, bq, cj, -bt,-bj, -cc, ca, bh, bv, -ch, -bo} {aj, ao, -ak, -ai, -an, al, ah, am, -am,-ah, -al, an, ai, ak, -ao, -aj, -aj, -ao, ak, ai, an, -al, - ah, -am,am, ah, al, -an, -ai, -ak, ao, aj, aj, ao, -ak, -ai, -an, al, ah, am,-am, -ah, -al, an, ai, ak, -ao, -aj, -aj, -ao, ak, ai, an, -al, -ah,-am, am, ah, al, -an, -ai, -ak, ao, aj} {bp, ck, -bq, -bo, -cj, br, bn,ci, -bs, -bm, -ch, bt, bl, cg, -bu, -bk, -cf, bv, bj, ce, -bw, -bi, -cd, bx, bh, cc, -by, -bg, -cb, bz, bf, ca, -ca, -bf, -bz, cb, bg, by,-cc, -bh, -bx, cd, bi, bw, - ce, -bj, -bv, cf, bk, bu, -cg, -bl, -bt,ch, bm, bs, -ci, -bn, -br, cj, bo, bq, -ck, -bp} {au, -be, -at, -av, bd,as, aw, -bc, -ar, -ax, bb, aq, ay, -ba, -ap, -az, az, ap, ba, -ay, -aq,-bb, ax, ar, bc, -aw, -as, -bd, av, at, be, -au, -au, be, at, av, -bd,-as, -aw, bc, ar, ax, -bb, -aq, -ay, ba, ap, az, -az, -ap, -ba, ay, aq,bb, -ax, -ar, -bc, aw, as, bd, -av, -at, -be, au} {bq, -ci, -bl, -bv,cd, bg, ca, -by, -bi, -cf, bt, bn, ck, -bo, -bs, cg, bj, bx, -cb, -bf,-cc, bw, bk, ch, -br, -bp, cj, bm, bu, -ce, -bh, -bz, bz, bh, ce, -bu,-bm, -cj, bp, br, -ch, -bk, -bw, cc, bf, cb, -bx, -bj, -cg, bs, bo, -ck,-bn, -bt, cf, bi, by, -ca, -bg, -cd, bv, bl, ci, -bq} {ae, -ag, -ad,-af, af, ad, ag, -ae, -ae, ag, ad, af, -af, -ad, -ag, ae, ae, -ag, -ad,-af, af, ad, ag, -ae, -ae, ag, ad, af, -af, -ad, -ag, ae, ae, -ag, -ad,-af, af, ad, ag, -ae, -ae, ag, ad, af, -af, - ad, -ag, ae, ae, -ag, -ad,-af, af, ad, ag, -ae, -ae, ag, ad, af, -af, -ad, -ag, ae} {br, -cf, -bg,-cc, bu, bo, -ci, -bj, -bz, bx, bl, ck, -bm, -bw, ca, bi, ch, -bp, -bt,cd, bf, ce, - bs, -bq, cg, bh, cb, -bv, -bn, cj, bk, by, -by, -bk, -cj,bn, bv, -cb, -bh, -cg, bq, bs, -ce, -bf, - cd, bt, bp, -ch, -bi, -ca,bw, bm, -ck, -bl, -bx, bz, bj, ci, -bo, -bu, cc, bg, cf, -br} {av, -bb,-ap, -bc, au, aw, -ba, -aq, -bd, at, ax, -az, -ar, -be, as, ay, -ay,-as, be, ar, az, -ax, -at, bd, aq, ba, -aw, -au, bc, ap, bb, -av, -av,bb, ap, bc, -au, -aw, ba, aq, bd, -at, -ax, az, ar, be, -as, -ay, ay,as, -be, -ar, -az, ax, at, -bd, -aq, -ba, aw, au, -bc, -ap, -bb, av}{bs, -cc, -bi, -cj, bl, bz, -bv, -bp, cf, bf, cg, -bo, -bw, by, bm, -ci,-bh, -cd, br, bt, -cb, -bj, - ck, bk, ca, -bu, -bq, ce, bg, ch, -bn,-bx, bx, bn, -ch, -bg, -ce, bq, bu, -ca, -bk, ck, bj, cb, - bt, -br, cd,bh, ci, -bm, -by, bw, bo, -cg, -bf, -cf, bp, bv, -bz, -bl, cj, bi, cc,-bs} {ak, -am, -ai, ao, ah, an, -aj, -al, al, aj, -an, -ah, -ao, ai, am,-ak, -ak, am, ai, -ao, -ah, -an, aj, al, -al, -aj, an, ah, ao, -ai, -am,ak, ak, -am, -ai, ao, ah, an, -aj, -al, al, aj, -an, -ah, -ao, ai, am,-ak, -ak, am, ai, -ao, -ah, -an, aj, al, -al, -aj, an, ah, ao, -ai, -am,ak} {bt, -bz, -bn, cf, bh, ck, -bi, -ce, bo, by, -bu, -bs, ca, bm, -cg,-bg, -cj, bj, cd, -bp, -bx, bv, br, -cb, -bl, ch, bf, ci, -bk, -cc, bq,bw, -bw, -bq, cc, bk, -ci, -bf, -ch, bl, cb, -br, -bv, bx, bp, -cd, -bj,cj, bg, cg, -bm, -ca, bs, bu, -by, -bo, ce, bi, -ck, -bh, -cf, bn, bz,-bt} {aw, -ay, -au, ba, as, -bc, -aq, be, ap, bd, -ar, -bb, at, az, -av,-ax, ax, av, -az, -at, bb, ar, - bd, -ap, -be, aq, bc, -as, -ba, au, ay,-aw, -aw, ay, au, -ba, -as, bc, aq, -be, -ap, -bd, ar, bb, - at, -az,av, ax, -ax, -av, az, at, -bb, -ar, bd, ap, be, -aq, -bc, as, ba, -au,-ay, aw} {bu, -bw, -bs, by, bq, -ca, -bo, cc, bm, -ce, -bk, cg, bi, -ci,-bg, ck, bf, cj, -bh, -ch, bj, cf, -bl, -cd, bn, cb, -bp, -bz, br, bx,-bt, -bv, bv, bt, -bx, -br, bz, bp, -cb, -bn, cd, bl, -cf, -bj, ch, bh,-cj, -bf, -ck, bg, ci, -bi, -cg, bk, ce, -bm, -cc, bo, ca, -bq, -by, bs,bw, -bu} {aa, -aa, -aa, aa, aa, -aa, -aa, aa, aa, -aa, -aa, aa, aa, -aa,-aa, aa, aa, -aa, -aa, aa, aa, -aa, - aa, aa, aa, -aa, -aa, aa, aa, -aa,-aa, aa, aa, -aa, -aa, aa, aa, -aa, -aa, aa, aa, -aa, -aa, aa, aa, - aa,-aa, aa, aa, -aa, -aa, aa, aa, -aa, -aa, aa, aa, -aa, -aa, aa, aa, -aa,-aa, aa} {bv, -bt, -bx, br, bz, -bp, -cb, bn, cd, -bl, -cf, bj, ch, -bh,-cj, bf, -ck, -bg, ci, bi, -cg, -bk, ce, bm, -cc, -bo, ca, bq, -by, -bs,bw, bu, -bu, -bw, bs, by, -bq, -ca, bo, cc, -bm, -ce, bk, cg, -bi, -ci,bg, ck, -bf, cj, bh, -ch, -bj, cf, bl, -cd, -bn, cb, bp, -bz, -br, bx,bt, -bv} {ax, -av, -az, at, bb, -ar, -bd, ap, -be, -aq, bc, as, -ba,-au, ay, aw, -aw, -ay, au, ba, -as, -bc, aq, be, -ap, bd, ar, -bb, -at,az, av, -ax, -ax, av, az, -at, -bb, ar, bd, -ap, be, aq, -bc, -as, ba,au, -ay, -aw, aw, ay, -au, -ba, as, bc, -aq, -be, ap, -bd, -ar, bb, at,-az, -av, ax} {bw, -bq, -cc, bk, ci, -bf, ch, bl, -cb, -br, bv, bx, -bp,-cd, bj, cj, -bg, cg, bm, -ca, -bs, bu, by, -bo, -ce, bi, ck, -bh, cf,bn, -bz, -bt, bt, bz, -bn, -cf, bh, -ck, -bi, ce, bo, -by, -bu, bs, ca,-bm, -cg, bg, -cj, -bj, cd, bp, -bx, -bv, br, cb, -bl, -ch, bf, -ci,-bk, cc, bq, -bw} {al, -aj, -an, ah, -ao, -ai, am, ak, -ak, -am, ai, ao,-ah, an, aj, -al, -al, aj, an, -ah, ao, ai, - am, -ak, ak, am, -ai, -ao,ah, -an, -aj, al, al, -aj, -an, ah, -ao, -ai, am, ak, -ak, -am, ai,ao, - ah, an, aj, -al, -al, aj, an, -ah, ao, ai, -am, -ak, ak, am, -ai,-ao, ah, -an, -aj, al} {bx, -bn, -ch, bg, -ce, -bq, bu, ca, -bk, -ck,bj, -cb, -bt, br, cd, -bh, ci, bm, -by, -bw, bo, cg, -bf, cf, bp, -bv,-bz, bl, cj, -bi, cc, bs, -bs, -cc, bi, -cj, -bl, bz, bv, -bp, -cf, bf,-cg, -bo, bw, by, -bm, -ci, bh, -cd, -br, bt, cb, -bj, ck, bk, -ca, -bu,bq, ce, -bg, ch, bn, -bx} {ay, -as, -be, ar, -az, -ax, at, bd, -aq, ba,aw, -au, -bc, ap, -bb, -av, av, bb, -ap, bc, au, -aw, -ba, aq, -bd, -at,ax, az, -ar, be, as, -ay, -ay, as, be, -ar, az, ax, -at, -bd, aq, -ba,-aw, au, bc, -ap, bb, av, -av, -bb, ap, -bc, -au, aw, ba, -aq, bd, at,-ax, -az, ar, -be, -as, ay} {by, -bk, cj, bn, -bv, -cb, bh, -cg, -bq,bs, ce, -bf, cd, bt, -bp, -ch, bi, -ca, -bw, bm, ck, -bl, bx, bz, -bj,ci, bo, -bu, -cc, bg, -cf, -br, br, cf, -bg, cc, bu, -bo, -ci, bj, -bz,-bx, bl, -ck, -bm, bw, ca, -bi, ch, bp, -bt, -cd, bf, -ce, -bs, bq, cg,-bh, cb, bv, -bn, -cj, bk, -by} {af, -ad, ag, ae, -ae, -ag, ad, -af,-af, ad, -ag, -ae, ae, ag, -ad, af, af, -ad, ag, ae, -ae, -ag, ad, -af,-af, ad, -ag, -ae, ae, ag, -ad, af, af, -ad, ag, ae, -ae, -ag, ad, -af,-af, ad, -ag, -ae, ae, ag, -ad, af, af, -ad, ag, ae, -ae, -ag, ad, -af,-af, ad, -ag, -ae, ae, ag, -ad, af} {bz, -bh, ce, bu, -bm, cj, bp, -br,-ch, bk, -bw, -cc, bf, -cb, -bx, bj, -cg, -bs, bo, ck, -bn, bt, cf, -bi,by, ca, -bg, cd, bv, -bl, ci, bq, -bq, -ci, bl, -bv, -cd, bg, -ca, -by,bi, -cf, -bt, bn, -ck, - bo, bs, cg, -bj, bx, cb, -bf, cc, bw, -bk, ch,br, -bp, -cj, bm, -bu, -ce, bh, -bz} {az, -ap, ba, ay, -aq, bb, ax, -ar,bc, aw, -as, bd, av, -at, be, au, -au, -be, at, -av, -bd, as, - aw, -bc,ar, -ax, -bb, aq, -ay, -ba, ap, -az, -az, ap, -ba, -ay, aq, -bb, -ax,ar, -be, -aw, as, -bd, - av, at, -be, -au, au, be, -at, av, bd, -as, aw,be, -ar, ax, bb, -aq, ay, ba, -ap, az} {ca, -bf, bz, cb, -bg, by, cc,-bh, bx, cd, -bi, bw, ce, -bj, bv, cf, -bk, bu, cg, -bl, bt, ch, - bm,bs, ci, -bn, br, cj, -bo, bq, ck, -bp, bp, -ck, -bq, bo, -cj, -br, bn,-ci, -bs, bm, -ch, -bt, bl, -cg, -bu, bk, -cf, -bv, bj, -ce, -bw, bi,-cd, -bx, bh, -cc, -by, bg, -cb, -bz, bf, -ca} {am, -ah, al, an, -ai,ak, ao, -aj, aj, -ao, -ak, ai, -an, -al, ah, -am, -am, ah, -al, -an, ai,-ak, - ao, aj, -aj, ao, ak, -ai, an, al, -ah, am, am, -ah, al, an, -ai,ak, ao, -aj, aj, -ao, -ak, ai, -an, - al, ah, -am, -am, ah, -al, -an,ai, -ak, -ao, aj, -aj, ao, ak, -ai, an, al, -ah, am} {cb, -bi, bu, ci,-bp, bn, -cg, -bw, bg, -bz, -cd, bk, -bs, -ck, br, -bl, ce, by, -bf, bx,cf, -bm, bq, -cj, -bt, bj, -cc, -ca, bh, -bv, -ch, bo, -bo, ch, bv, -bh,ca, cc, -bj, bt, cj, -bq, bm, -cf, -bx, bf, -by, -ce, bl, -br, ck, bs,-bk, cd, bz, -bg, bw, cg, -bn, bp, -ci, -bu, bi, -cb} {ba, -ar, av, -be,-aw, aq, -az, -bb, as, -au, bd, ax, -ap, ay, bc, -at, at, -bc, -ay, ap,-ax, -bd, au, -as, bb, az, -aq, aw, be, -av, ar, -ba, -ba, ar, -av, be,aw, -aq, az, bb, -as, au, -bd, -ax, ap, -ay, -bc, at, -at, bc, ay, -ap,ax, bd, -au, as, -bb, -az, aq, -aw, -be, av, -ar, ba} {cc, -bl, bp, -cg,-by, bh, -bt, ck, bu, -bg, bx, ch, -bq, bk, -cb, -cd, bm, -bo, cf, bz,-bi, bs, -cj, -bv, bf, -bw, -ci, br, -bj, ca, ce, -bn, bn, -ce, -ca, bj,-br, ci, bw, -bf, bv, cj, -bs, bi, -bz, - cf, bo, -bm, cd, cb, -bk, bq,-ch, -bx, bg, -bu, -ck, bt, -bh, by, cg, -bp, bl, -cc} {ac, -ab, ab,-ac, -ac, ab, -ab, ac, ac, -ab, ab, -ac, -ac, ab, -ab, ac, ac, -ab, ab,-ac, -ac, ab, - ab, ac, ac, -ab, ab, -ac, -ac, ab, -ab, ac, ac, -ab, ab,-ac, -ac, ab, -ab, ac, ac, -ab, ab, -ac, -ac, ab, -ab, ac, ac, -ab, ab,-ac, -ac, ab, -ab, ac, ac, -ab, ab, -ac, -ac, ab, -ab, ac} {cd, -bo, bk,-bz, -ch, bs, -bg, bv, -ck, -bw, bh, -br, cg, ca, -bl, bn, -cc, -ce, bp,-bj, by, ci, -bt, bf, -bu, cj, bx, -bi, bq, -cf, -cb, bm, -bm, cb, cf,-bq, bi, -bx, -cj, bu, -bf, bt, -ci, -by, bj, -bp, ce, cc, -bn, bl, -ca,-cg, br, -bh, bw, ck, -bv, bg, -bs, ch, bz, -bk, bo, -cd} {bb, -au, aq,-ax, be, ay, -ar, at, -ba, -bc, av, -ap, aw, -bd, -az, as, -as, az, bd,-aw, ap, -av, bc, ba, -at, ar, -ay, -be, ax, -aq, au, -bb, -bb, au, -aq,ax, -be, -ay, ar, -at, ba, bc, -av, ap, - aw, bd, az, -as, as, -az, -bd,aw, -ap, av, -bc, -ba, at, -ar, ay, be, -ax, aq, -au, bb} {ce, -br, bf,-bs, cf, cd, -bq, bg, -bt, cg, cc, -bp, bh, -bu, ch, cb, -bo, bi, -bv,ci, ca, -bn, bj, -bw, cj, bz, -bm, bk, -bx, ck, by, -bl, bl, -by, -ck,bx, -bk, bm, -bz, -cj, bw, -bj, bn, -ca, - ci, bv, -bi, bo, -cb, -ch,bu, -bh, bp, -cc, -cg, bt, -bg, bq, -cd, -cf, bs, -bf, br, -ce} {an,-ak, ah, -aj, am, ao, -al, ai, -ai, al, -ao, -am, aj, -ah, ak, -an, -an,ak, -ah, aj, -am, -ao, al, -ai, ai, -al, ao, am, -aj, ah, -ak, an, an,-ak, ah, -aj, am, ao, -al, ai, -ai, al, -ao, -am, aj, - ah, ak, -an,-an, ak, -ah, aj, -am, -ao, al, -ai, ai, -al, ao, am, -aj, ah, -ak, an}{cf, -bu, bj, -bl, bw, -ch, -cd, bs, -bh, bn, -by, cj, cb, -bq, bf, -bp,ca, ck, -bz, bo, -bg, br, -cc, -ci, bx, -bm, bi, -bt, ce, cg, -bv, bk,-bk, bv, -cg, -ce, bt, -bi, bm, -bx, ci, cc, -br, bg, - bo, bz, -ck,-ca, bp, -bf, bq, -cb, -cj, by, -bn, bh, -bs, cd, ch, -bw, bl, -bj, bu,-cf} {bc, -ax, as, -aq, av, -ba, -be, az, -au, ap, -at, ay, -bd, -bb,aw, -ar, ar, -aw, bb, bd, -ay, at, -ap, au, -az, be, ba, -av, aq, -as,ax, -bc, -bc, ax, -as, aq, -av, ba, be, -az, au, -ap, at, -ay, bd, bb,-aw, ar, -ar, aw, -bb, -bd, ay, -at, ap, -au, az, -be, -ba, av, -aq, as,-ax, bc} {cg, -bx, bo, -bf, bn, -bw, cf, ch, -by, bp, -bg, bm, -bv, ce,ci, -bz, bq, -bh, bl, -bu, cd, cj, -ca, br, -bi, bk, -bt, cc, ck, -cb,bs, -bj, bj, -bs, cb, -ck, -cc, bt, -bk, bi, -br, ca, -cj, -cd, bu, -bl, bh, -bq, bz, -ci, -ce, bv, -bm, bg, -bp, by, -ch, -cf, bw, -bn, bf,-bo, bx, -cg} {ag, -af, ae, -ad, ad, -ae, af, -ag, -ag, af, -ae, ad,-ad, ae, -af, ag, ag, -af, ae, -ad, ad, -ae, af, -ag, -ag, af, -ae, ad,-ad, ae, -af, ag, ag, -af, ae, -ad, ad, -ae, af, -ag, -ag, af, -ae, ad,-ad, ae, -af, ag, ag, -af, ae, -ad, ad, -ae, af, -ag, -ag, af, -ae, ad,-ad, ae, -af, ag} {ch, -ca, bt, -bm, bf, -bl, bs, -bz, cg, ci, -cb, bu,-bn, bg, -bk, br, -by, cf, cj, -cc, bv, -bo, bh, -bj, bq, -bx, ce, ck,-cd, bw, -bp, bi, -bi, bp, -bw, cd, -ck, -ce, bx, -bq, bj, -bh, bo, -bv,cc, -cj, -cf, by, -br, bk, -bg, bn, -bu, cb, -ci, -cg, bz, -bs, bl, -bf,bm, -bt, ca, -ch} {bd, -ba, ax, -au, ar, -ap, as, -av, ay, -bb, be, be,-az, aw, -at, aq, -aq, at, -aw, az, -bc, -be, bb, -ay, av, -as, ap, -ar,au, -ax, ba, -bd, -bd, ba, -ax, au, -ar, ap, -as, av, -ay, bb, -be, -bc,az, -aw, at, -aq, aq, -at, aw, -az, bc, be, -bb, ay, -av, as, -ap, ar,-au, ax, -ba, bd} {ci, -cd, by, -bt, bo, -bj, bf, -bk, bp, -bu, bz, -ce,cj, ch, -cc, bx, -bs, bn, -bi, bg, -bl, bq, - bv, ca, -cf, ck, cg, -cb,bw, -br, bm, -bh, bh, -bm, br, -bw, cb, -cg, -ck, cf, -ca, bv, -bq,bl, - bg, bi, -bn, bs, -bx, cc, -ch, -cj, ce, -bz, bu, -bp, bk, -bf, bj,-bo, bt, -by, cd, -ci} {ao, -an, am, -al, ak, -aj, ai, -ah, ah, -ai, aj,-ak, al, -am, an, -ao, -ao, an, -am, al, -ak, aj, - ai, ah, -ah, ai,-aj, ak, -al, am, -an, ao, ao, -an, am, -al, ak, -aj, ai, -ah, ah, -ai,aj, -ak, al, - am, an, -ao, -ao, an, -am, al, -ak, aj, -ai, ah, -ah, ai,-aj, ak, -al, am, -an, ao} {cj, -cg, cd, -ca, bx, -bu, br, -bo, bl, -bi,bf, -bh, bk, -bn, bq, -bt, bw, -bz, cc, -cf, ci, ck, - ch, ce, -cb, by,-bv, bs, -bp, bm, -bj, bg, -bg, bj, -bm, bp, -bs, bv, -by, cb, -ce, ch,-ck, -ci, cf, -cc, bz, -bw, bt, -bq, bn, -bk, bh, -bf, bi, -bl, bo, -br,bu, -bx, ca, -cd, cg, -cj} {be, -bd, bc, -bb, ba, -az, ay, -ax, aw, -av,au, -at, as, -ar, aq, -ap, ap, -aq, ar, -as, at, -au, av, -aw, ax, -ay,az, -ba, bb, -bc, bd, -be, -be, bd, -bc, bb, -ba, az, -ay, ax, -aw, av,-au, at, - as, ar, -aq, ap, -ap, aq, -ar, as, -at, au, -av, aw, -ax, ay,-az, ba, -bb, bc, -bd, be} {ck, -cj, ci, -ch, cg, -cf, ce, -cd, cc, -cb,ca, -bz, by, -bx, bw, -bv, bu, -bt, bs, -br, bq, -bp, bo, -bn, bm, -bl,bk, -bj, bi, -bh, bg, -bf, bf, -bg, bh, -bi, bj, -bk, bl, -bm, bn, -bo,bp, -bq, br, -bs, bt, -bu, bv, -bw, bx, -by, bz, -ca, cb, -cc, cd, -ce,cf, -cg, ch, -ci, cj, -ck} } where {aa, ab, ac, ad, ae, af, ag, ah, ai,aj, ak, al, am, an, ao, ap, aq, ar, as, at, au, av, aw, ax, ay, az, ba,bb, bc, bd, be, bf, bg, bh, bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs,bt, bu, bv, bw, bx, by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck}={64, 83, 36, 89, 75, 50, 18, 90, 87, 80, 70, 57, 43, 25, 9, 90, 90, 88,85, 82, 78, 73, 67, 61, 54, 46, 38, 31, 22, 13, 4, 9 1, 90, 90, 90, 88,87, 86, 84, 83, 81, 79, 77, 73, 71, 69, 65, 62, 59, 56, 52, 48, 44, 41,37, 33, 28, 24, 20, 15, 11, 7, 2}

Appendix III

4-point DST-7

{ a, b, c, d} { c, c, 0, −c} { d, −a, −c, b} { b, −d, c, −a} where {a,b, c, d} = { 29, 55, 74, 84}

8-point DST-7:

{ a, b, c, d, e, f, g, h,} { c, f, h, e, b, −a, −d, −g,} { e, g, b, −c,−h, −d, a, f,} { g, c, −d, −f, a, h, b, −e,} { h, −a, −g, b, f, −c, −e,d,} { f, −e, −a, g, −d, −b, h, −c,} { d, −h, e, −a, −c, g, −f, b,} { b,−d, f, −h, g, −e, c, −a,} where {a, b, c, d, e, f, g, h} = { 17, 32, 46,60, 71, 78, 85, 86}

16-point DST-7

{ a, b, c, d, e, f, g, h, i, j, k, 1, m, n, o, p,} { c, f, i, 1, o, o,1, i, f, c, 0, −c, −f, −i, −1, −o,} { e, j, o, m, h, c, −b, −g, −1, −p,−k, −f, −a, d, i, n,} { g, n, 1, e, −b, −i, −p, −j, −c, d, k, o, h, a,−f, −m,} { i, o, f, −c, −1, −1, −c, f, o, i, 0, −i, −o, −f, c, 1,} { k,k, 0, −k, −k, 0, k, k, 0, −k, −k, 0, k, k, 0, −k,} { m, g, −f, −n, −a,1, h, −e, −o, −b, k, i, −d, −p, −c, j,} { o, c, −1, −f, i, i, −f, −1, c,o, 0, −o, −c, 1, f, −i,} { p, −a, −o, b, n, −c, −m, d, 1, −e, −k, f, j,−g, −i, h,} { n, −e, −i, j, d, −o, a, m, −f, −h, k, c, −p, b, 1, −g,} {1, −i, −c, o, −f, −f, o, −c, −i, 1, 0, −1, i, c, −o, f,} { j, −m, c, g,−p, f, d, −n, i, a, −k, 1, −b, −h, o, −e,} { h, −p, i, −a, −g, o, −j, b,f, −n, k, −c, −e, m, −1, d,} { f, −1, o, −i, c, c, −i, o, −1, f, 0, −f,1, −o, i, −c,} { d, −h, 1, −p, m, −i, e, −a, −c, g, −k, o, −n, j, −f,b,} { b, −d, f, −h, j, −1, n, −p, o, −m, k, −i, g, −e, c, −a,} where {a,b, c, d, e, f, g, h, i, j, k, 1, m, n, o, p} = { 9, 17, 25, 33, 41, 49,56, 62, 66, 72, 77, 81, 83, 87, 89, 90}

32-point DST-7

{ a, b, c, d, e, f, g, h, i, j, k, 1, m, n, o, p, q, r, s, t, u, v, w,x, y, z, A, B, C, D, E, F,} { c, f, i, 1, o, r, u, x, A, D, F, C, z, w,t, q, n, k, h, e, b, −a, −d, −g, −j, −m, −p, −s, −v, −y, −B, −E,} { e,j, o, t, y, D, D, y, t, o, j, e, 0, −e, −j, −o, −t, −y, −D, −D, −y, −t,−o, −j, −e, 0, e, j, o, t, y, D,} { g, n, u, B, D, w, p, i, b, −e, −1,−s, −z, −F, −y, −r, −k, −d, c, j, q, x, E, A, t, m, f, −a, −h, −o, −v,−C,} { i, r, A, C, t, k, b, −g, −p, −y, −E, −v, −m, −d, e, n, w, F, x,o, f, −c, −1, −u, −D, −z, −q, −h, a, j, s, B,} { k, v, F, u, j, −a, −1,−w, −E, −t, −i, b, m, x, D, s, h, −c, −n, −y, −C, −r, −g, d, o, z, B, q,f, −e, −p, −A,} { m, z, z, m, 0, −m, −z, −z, −m, 0, m, z, z, m, 0, −m,−z, −z, −m, 0, m, z, z, m, 0, −m, −z, −z, −m, 0, m, z,} { o, D, t, e,−j, −y, −y, −j, e, t, D, o, 0, −o, −D, −t, −e, j, y, y, j, −e, −t, −D,−o, 0, o, D, t, e, −j, −y, } { q, E, n, −c, −t, −B, −k, f, w, y, h, −i,−z, −v, −e, 1, C, s, b, −o, −F, −p, a, r, D, m, −d, −u, −A, −j, g, x, }{ s, A, h, −k, −D, −p, c, v, x, e, −n, −F, −m, f, y, u, b, −q, −C, −j,i, B, r, −a, −t, −z, −g, 1, E, o, −d, −w,} { u, w, b, −s, −y, −d, q, A,f, −o, −C, −h, m, E, j, −k, −F, −1, i, D, n, −g, −B, −p, e, z, r, −c,−x, −t, a, v,} { w, s, −d, −A, −o, h, E, k, −1, −D, −g, p, z, c, −t, −v,a, x, r, −e, −B, −n, i, F, j, −m, −C, −f, q, y, b, −u, } { y, o, −j, −D,−e, t, t, −e, −D, −j, o, y, 0, −y, −o, j, D, e, −t, −t, e, D, j, −o, −y,0, y, o, −j, −D, −e, t,} { A, k, −p, −v, e, F, f, −u, −q, j, B, a, −z,−1, o, w, −d, −E, −g, t, r, −i, −C, −b, y, m, −n, −x, c, D, h, −s, } {C, g, −v, −n, o, u, −h, −B, a, D, f, −w, −m, p, t, −i, −A, b, E, e, −x,−1, q, s, −j, −z, c, F, d, −y, −k, r,} { E, c, −B, −f, y, i, −v, −1, s,o, −p, −r, m, u, −j, −x, g, A, −d, −D, a, F, b, −C, −e, z, h, −w, −k, t,n, −q,} { F, −a, −E, b, D, −c, −C, d, B, −e, −A, f, z, −g, −y, h, x, −i,−w, j, v, −k, −u, 1, t, −m, −s, n, r, −o, −q, p,} { D, −e, −y, j, t, −o,−o, t, j, −y, −e, D, 0, −D, e, y, −j, −t, o, o, −t, −j, y, e, −D, 0, D,−e, −y, j, t, −o,} { B, −i, −s, r, j, −A, −a, C, −h, −t, q, k, −z, −b,D, −g, −u, p, 1, −y, −c, E, −f, −v, o, m, −x, −d, F, −e, −w, n,} { z,−m, −m, z, 0, −z, m, m, −z, 0, z, −m, −m, z, 0, −z, m, m, −z, 0, z, −m,−m, z, 0, −z, m, m, −z, 0, z, −m,} { x, −q, −g, E, −j, −n, A, −c, −u, t,d, −B, m, k, −D, f, r, −w, −a, y, −p, −h, F, −i, −o, z, −b, −v, s, e,−C, 1,} { v, −u, −a, w, −t, −b, x, −s, −c, y, −r, −d, z, −q, −e, A, −p,−f, B, −o, −g, C, −n, −h, D, −m, −i, E, −1, −j, F, −k, } { t, −y, e, o,−D, j, j, −D, o, e, −y, t, 0, −t, y, −e, −o, D, −j, −j, D, −o, −e, y,−t, 0, t, −y, e, o, −D, j,} { r, −C, k, g, −y, v, −d, −n, F, −o, −c, u,−z, h, j, −B, s, −a, −q, D, −1, −f, x, −w, e, m, −E, p, b, −t, A, −i,} {p, −F, q, −a, −o, E, −r, b, n, −D, s, −c, −m, C, −t, d, 1, −B, u, −e,−k, A, −v, f, j, −z, w, −g, −i, y, −x, h,} { n, −B, w, −i, −e, s, −F, r,−d, −j, x, −A, m, a, −o, C, −v, h, f, −t, E, −q, c, k, −y, z, −1, −b, p,−D, u, −g,} { 1, −x, C, −q, e, g, −s, E, −v, j, b, −n, z, −A, o, −c, −i,u, −F, t, −h, −d, p, −B, y, −m, a, k, −w, D, −r, f,} { j, −t, D, −y, o,−e, −e, o, −y, D, −t, j, 0, −j, t, −D, y, −o, e, e, −o, y, −D, t, −j, 0,j, −t, D, −y, o, −e,} { h, −p, x, −F, y, −q, i, −a, −g, o, −w, E, −z, r,−j, b, f, −n, v, −D, A, −s, k, −c, −e, m, −u, C, −B, t, −1, d,} { f, −1,r, −x, D, −C, w, −q, k, −e, −a, g, −m, s, −y, E, −B, v, −p, j, −d, −b,h, −n, t, −z, F, −A, u, −o, i, −c,} { d, −h, 1, −p, t, −x, B, −F, C, −y,u, −q, m, −i, e, −a, −c, g, −k, o, −s, w, −A, E, −D, z, −v, r, −n, j,−f, b,} { b, −d, f, −h, j, −1, n, −p, r, −t, v, −x, z, −B, D, −F, E, −C,A, −y, w, −u, s, −q, o, −m, k, −i, g, −e, c, −a,} where {a, b, c, d, e,f, g, h, i, j, k, 1, m, n, o, p, q, r, s, t, u, v, w, x, y, z, A, B, C,D, E, F } = {4, 9, 13, 17, 21, 26, 30, 34, 38, 42, 45, 50, 53, 56, 60,63, 66, 68, 72, 74, 77, 78, 80, 82, 84, 85, 86, 88, 88, 89, 90, 90}

4-point DCT-8

{ a, b, c, d,} { b, 0, −b, −b,} { c, −b, −d, a,} { d, −b, a, −c,} where{a, b, c, d} = { 84, 74, 55, 29}

8-point DCT-8:

{ a, b, c, d, e, f, g, h,} { b, e, h, −g, −d, −a, −c, −f,} { c, h, −e,−a, −f, g, b, d,} { d, −g, −a, −h, c, e, −f, −b,} { e, −d, −f, c, g, −b,−h, a,} { f, −a, g, e, −b, h, d, −c,} { g, −c, b, −f, −h, d, −a, e,} {h, −f, d, −b, a, −c, e, −g,} where {a, b, c, d, e, f, g, h} = { 86, 85,78, 71, 60, 46, 32, 17}

16-point DCT-8

{ a, b, c, d, e, f, g, h, i, j, k, 1, m, n, o, p,} { b, e, h, k, n, 0,−n, −k, −h, −e, −b, −b, −e, −h, −k, −n,} { c, h, m, −p, −k, −f, −a, −e,−j, −o, n, i, d, b, g, 13 { d, k, −p, −i, −b, −f, −m, n, g, a, h, o, −1,−e, −c, −j,} { e, n, −k, −b, −h, 0, h, b, k, −n, −e, −e, −n, k, b, h,} {f, 0, −f, −f, 0, f, f, 0, −f, −f, 0, f, f, 0, −f, −f,} { g, −n, −a, −m,h, f, −o, −b, −1, i, e, −p, −c, −k, j, d,} { h, −k, −e, n, b, 0, −b, −n,e, k, −h, −h, k, e, −n, −b,} { i, −h, −j, g, k, −f, −1, e, m, −d, −n, c,o, −b, −p, a,} { j, −e, −o, a, −n, −f, i, k, −d, −p, b, −m, −g, h, 1,−c,} { k, −b, n, h, −e, 0, e, −h, −n, b, −k, −k, b, −n, −h, e,} { 1, −b,i, o, −e, f, −p, −h, c, −m, −k, a, −j, −n, d, −g,} { m, −e, d, −1, −n,f, −c, k, o, −g, b, −j, −p, h, −a, i,} { n, −h, b, −e, k, 0, −k, e, −b,h, −n, −n, h, −b, e, −k,} { o, −k, g, −c, b, −f, j, −n, −p, 1, −h, d,−a, e, −i, m,} { p, −n, 1, −j, h, −f, d, −b, a, −c, e, −g, i, −k, m,−o,} where {a, b, c, d, e, f, g, h, i, j, k, 1, m, n, o, p} = {90, 89,87, 83, 81, 77, 72, 66, 62, 56, 49, 41, 33, 25, 17, 9}

32-point DCT-8

{ a, b, c, d, e, f, g, h, i, j, k, 1, m, n, o, p, q, r, s, t, u, v, w,x, y, z, A, B, C, D, E, F,} { b, e, h, k, n, q, t, w, z, C, F, −E, −B,−y, −v, −s, −p, −m, −j, −g, −d, −a, −c, −f, −i, −1, −o, −r, −u, −x, −A,−D,} { c, h, m, r, w, B, 0, −B, −w, −r, −m, −h, −c, −c, −h, −m, −r, −w,−B, 0, B, w, r, m, h, c, c, h, m, r, w, B,} { d, k, r, y, F, −A, −t, −m,−f, −b, −i, −p, −w, −D, C, v, o, h, a, g, n, u, B, −E, −x, −q, −j, −c,−e, −1, −s, −z,} { e, n, w, F, −y, −p, −g, −c, −1, −u, −D, A, r, i, a,j, s, B, −C, −t, −k, −b, −h, −q, −z, E, v, m, d, f, o, x,} { f, q, B,−A, −p, −e, −g, −r, −C, z, o, d, h, s, D, −y, −n, −c, −i, −t, −E, x, m,b, j, u, F, −w, −1, −a, −k, −v,} { g, t, 0, −t, −g, −g, −t, 0, t, g, g,t, 0, −t, −g, −g, −t, 0, t, g, g, t, 0, −t, −g, −g, −t, 0, t, g, g, t,}{ h, w, −B, −m, −c, −r, 0, r, c, m, B, −w, −h, −h, −w, B, m, c, r, 0,−r, −c, −m, −B, w, h, h, w, −B, −m, −c, −r,} { i, z, −w, −f, −1, −C, t,c, o, F, −q, −a, −r, E, n, d, u, −B, −k, −g, −x, y, h, j, A, −v, −e, −m,−D, s, b, p,} { j, C, −r, −b, −u, z, g, m, F, −o, −e, −x, w, d, p, −E,−1, −h, −A, t, a, s, −B, −i, −k, −D, q, c, v, −y, −f, −n,} { k, F, −m,−i, −D, o, g, B, −q, −e, −z, s, c, x, −u, −a, −v, w, b, t, −y, −d, −r,A, f, p, −C, −h, −n, E, j, 1,} { 1, −E, −h, −p, A, d, t, −w, −a, −x, s,e, B, −o, −i, −F, k, m, −D, −g, −q, z, c, u, −v, −b, −y, r, f, C, −n,−j,} { m, −B, −c, −w, r, h, 0, −h, −r, w, c, B, −m, −m, B, c, w, −r, −h,0, h, r, −w, −c, −B, m, m, −B, −c, −w, r, h,} { n, −y, −c, −D, i, s, −t,−h, E, d, x, −o, −m, z, b, C, −j, −r, u, g, −F, −e, −w, p, 1, −A, −a,−B, k, q, −v, −f,} { o, −v, −h, C, a, D, −g, −w, n, p, −u, −i, B, b, E,−f, −x, m, q, −t, −j, A, c, F, −e, −y, 1, r, −s, −k, z, d,} { p, −s, −m,v, j, −y, −g, B, d, −E, −a, −F, c, C, −f, −z, i, w, −1, −t, o, q, −r,−n, u, k, −x, −h, A, e, −D, −b,} { q, −p, −r, o, s, −n, −t, m, u, −1,−v, k, w, −j, −x, i, y, −h, −z, g, A, −f, −B, e, C, −d, −D, c, E, −b,−F, a,} { r, −m, −w, h, B, −c, 0, c, −B, −h, w, m, −r, −r, m, w, −h, −B,c, 0, −c, B, h, −w, −m, r, r, −m, −w, h, B, −c,} { s, −j, −B, a, −C, −i,t, r, −k, −A, b, −D, −h, u, q, −1, −z, c, −E, −g, v, p, −m, −y, d, −F,−f, w, o, −n, −x, e,} { t, −g, 0, g, −t, −t, g, 0, −g, t, t, −g, 0, g,−t, −t, g, 0, −g, t, t, −g, 0, g, −t, −t, g, 0, −g, t, t, −g,} { u, −d,B, n, −k, −E, g, −r, −x, a, −y, −q, h, −F, −j, o, A, −c, v, t, −e, C, m,−1, −D, f, −s, −w, b, −z, −p, i,} { v, −a, w, u, −b, x, t, −c, y, s, −d,z, r, −e, A, q, −f, B, p, −g, C, o, −h, D, n, −i, E, m, −j, F, 1, −k,} {w, −c, r, B, −h, m, 0, −m, h, −B, −r, c, −w, −w, c, −r, −B, h, −m, 0, m,−h, B, r, −c, w, w, −c, r, B, −h, m,} { x, −f, m, −E, −q, b, −t, −B, j,−i, A, u, −c, p, F, −n, e, −w, −y, g, −1, D, r, −a, s, C, −k, h, −z, −v,d, −o,} { y, −i, h, −x, −z, j, −g, w, A, −k, f, −v, −B, 1, −e, u, C, −m,d, −t, −D, n, −c, s, E, −o, b, −r, −F, p, −a, q,} { z, −1, c, −q, E, u,−g, h, −v, −D, p, −b, m, −A, −y, k, −d, r, −F, −t, f, −i, w, C, −o, a,−n, B, x, −j, e, −s,} { A, −o, c, −j, v, F, −t, h, −e, q, −C, −y, m, −a,1, −x, −D, r, −f, g, −s, E, w, −k, b, −n, z, B, −p, d, −i, u,} { B, −r,h, −c, m, −w, 0, w, −m, c, −h, r, −B, −B, r, −h, c, −m, w, 0, −w, m, −c,h, −r, B, B, −r, h, −c, m, −w,} { C, −u, m, −e, d, −1, t, −B, −D, v, −n,f, −c, k, −s, A, E, −w, o, −g, b, −j, r, −z, −F, x, −p, h, −a, i, −q,y,} { D, −x, r, −1, f, −a, g, −m, s, −y, E, C, −w, q, −k, e, −b, h, −n,t, −z, F, B, −v, p, −j, d, −c, i, −o, u, −A,} { E, −A, w, −s, o, −k, g,−c, b, −f, j, −n, r, −v, z, −D, −F, B, −x, t, −p, 1, −h, d, −a, e, −i,m, −q, u, −y, C,} { F, −D, B, −z, x, −v, t, −r, p, −n, 1, −j, h, −f, d,−b, a, −c, e, −g, i, −k, m, −o, q, −s, u, −w, y, −A, C, −E,} where {a,b, c, d, e, f, g, h, i, j, k, 1, m, n, o, p, q, r, s, t, u, v, w, x, y,z, A, B, C, D, E, F } = {90, 90, 89, 88, 88, 86, 85, 84, 82, 80, 78, 77,74, 72, 68, 66, 63, 60, 56, 53, 50, 45, 42, 38, 34, 30, 26, 21, 17, 13,9, 4}

What is claimed is:
 1. A method for video decoding in a decoder,comprising: decoding coded information of a coding unit (CU) from acoded video bitstream, the coded information indicating a last positionof non-zero transform coefficients of a first coding block (CB) of theCU; determining whether a secondary transform index is signaled in thecoded information based on the last position; determining whether toperform a secondary transform on a second CB based on whether thesecondary transform index is determined to be signaled in the codedinformation; responsive to the secondary transform being determined tobe performed, performing the secondary transform on the second CB andreconstructing the second CB; and responsive to the secondary transformbeing determined not to be performed, reconstructing the second CBwithout performing the secondary transform on the second CB.
 2. Themethod of claim 1, wherein the determining whether the secondarytransform index is signaled further comprises: determining whether ahorizontal component of the last position is less then a first thresholdand a vertical component of the last position is less than a secondthreshold; and responsive to the horizontal component being determinedto be less than the first threshold and the vertical component beingdetermined to be less than the second threshold, determining that thesecondary transform index is not signaled in the coded information. 3.The method of claim 1, wherein the determining whether the secondarytransform index is signaled further comprises: determining whether a sumof a horizontal component and a vertical component of the last positionis less than a threshold; and responsive to the sum being determined tobe less than the threshold, determining that the secondary transformindex is not signaled in the coded information.
 4. The method of claim1, wherein the determining whether the secondary transform index issignaled further comprises: determining whether a minimum one of (i) ahorizontal component and (ii) a vertical component of the last positionis less than a threshold; and responsive to the minimum one beingdetermined to be less than the threshold, determining that the secondarytransform index is not signaled in the coded information.
 5. The methodof claim 1, wherein the determining whether the secondary transformindex is signaled further comprises: determining whether a maximum oneof (i) a horizontal component and (ii) a vertical component of the lastposition is less than a threshold; and responsive to the maximum onebeing determined to be less than the threshold, determining that thesecondary transform index is not signaled in the coded information. 6.The method of claim 1, wherein the first CB is a luma block; the lastposition is a last luma position for the luma block; and the determiningwhether the second transform index is signaled further includesdetermining whether the second transform index is signaled based on thelast luma position.
 7. The method of claim 1, wherein the first CB is aluma block; the last position is a last luma position for the lumablock; the CU further includes a chroma block; the coding informationfurther indicates a last chroma position of non-zero transformcoefficients for the chroma block; and the determining whether thesecondary transform index is signaled further includes determiningwhether the secondary transform index is signaled based on the last lumaposition and the last chroma position.
 8. A method for video decoding ina decoder, comprising: decoding coding information of a coding unit (CU)from a coded video bitstream, the coding information indicating a sizeof the CU; and determining whether a secondary transform is allowedbased on the size of the CU and a CU size threshold, wherein when thesize of the CU is less than or equal to the CU size threshold, thesecondary transform is determined to be allowed, and when the size ofthe CU is larger than the CU size threshold, the secondary transform isdetermined not to be allowed.
 9. The method of claim 8, wherein the CUsize threshold is a maximum size of a transform unit in the CU.
 10. Themethod of claim 8, further comprising: when the size of the CU is lessthan or equal to the CU size threshold, determining a number of non-zerotransform coefficients for at least one CB in the CU, a size of each ofthe at least one CB being larger than or equal to a first threshold; andresponsive to the number of non-zero transform coefficients being lessthan a second threshold, determining that a secondary transform index isnot signaled in the coded information.
 11. The method of claim 10,wherein the coding information indicates that a color format for the CUis YUV 4:2:0; the CU includes a luma block and two chroma blocks; andthe method further includes: determining whether a first dimension ofthe luma block is 4 and a second dimension of the luma block is N, Nbeing larger than or equal to 4; and responsive to the first and seconddimensions being determined to be 4 and N, respectively, the determiningthe number of non-zero transform coefficients includes determining thenumber of non-zero transform coefficients from only the luma block, theat least one CB being the luma block.
 12. The method of claim 10,wherein the coding information indicates that a color format for the CUis YUV 4:2:2; the CU includes a luma block and two chroma blocks; andthe method further includes: determining whether a size of the lumablock is 4×N, N being larger than or equal to 4; and responsive to thesize of the luma block being determined to be 4×N, N and 4 being aheight and a width of the luma block, respectively, the determining thenumber of non-zero transform coefficients includes determining thenumber of non-zero transform coefficients from only the luma block, theat least one CB being the luma block.
 13. An apparatus for videodecoding, comprising processing circuitry configured to: decode codedinformation of a coding unit (CU) from a coded video bitstream, thecoded information indicating a last position of non-zero transformcoefficients of a first coding block (CB) of the CU; determine whether asecondary transform index is signaled in the coded information based onthe last position; determine whether to perform a secondary transform ona second CB based on whether the secondary transform index is determinedto be signaled in the coded information; responsive to the secondarytransform being determined to be performed, performing the secondarytransform on the second CB and reconstructing the second CB; andresponsive to the secondary transform being determined not to beperformed, reconstruct the second CB without performing the secondarytransform on the second CB.
 14. The apparatus of claim 13, wherein theprocessing circuitry is further configured to: determine whether ahorizontal component of the last position is less then a first thresholdand a vertical component of the last position is less than a secondthreshold; and responsive to the horizontal component being determinedto be less than the first threshold and the vertical component beingdetermined to be less than the second threshold, determine that thesecondary transform index is not signaled in the coded information. 15.The apparatus of claim 13, wherein the processing circuitry is furtherconfigured to: determine whether a sum of a horizontal component and avertical component of the last position is less than a threshold; andresponsive to the sum being determined to be less than the threshold,determine that the secondary transform index is not signaled in thecoded information.
 16. The apparatus of claim 13, wherein the processingcircuitry is further configured to: determine whether a minimum one or amaximum one of (i) a horizontal component and (ii) a vertical componentof the last position is less than a threshold; and responsive to one ofthe minimum one and the maximum one being determined to be less than thethreshold, determine that the secondary transform index is not signaledin the coded information.
 17. The apparatus of claim 13, wherein thefirst CB is a luma block; the last position is a last luma position forthe luma block; and the processing circuitry is further configured todetermine whether the second transform index is signaled based on thelast luma position.
 18. The apparatus of claim 13, wherein the first CBis a luma block; the last position is a last luma position for the lumablock; the CU further includes a chroma block; the coding informationfurther indicates a last chroma position of non-zero transformcoefficients for the chroma block; and the processing circuitry isfurther configured to determine whether the secondary transform index issignaled based on the last luma position and the last chroma position.19. An apparatus for video decoding, comprising processing circuitryconfigured to: decode coding information of a coding unit (CU) from acoded video bitstream, the coding information indicating a size of theCU; and determine whether a secondary transform is allowed based on thesize of the CU and a CU size threshold, wherein when the size of the CUis less than or equal to the CU size threshold, the secondary transformis determined to be allowed, and when the size of the CU is larger thanthe CU size threshold, the secondary transform is determined not to beallowed.
 20. The apparatus of claim 19, wherein the CU size threshold isa maximum size of a transform unit in the CU.